2.1 Basic properties

We study the basic properties of the spaces $X_G$ and $[X_G/G]$. Some of the results in this section were also obtained, in the more general situation of stacks of L-parameters, by Dat, Helm, Kurinczuk and Moss [Reference Dat, Helm, Kurinczuk and MossDHKM20] and Zhu [Reference ZhuZhu20].

Example 2.3 Let $r,d>0$ and $n=rd$. Let $B\subset \operatorname {GL}_n$ be the Borel subgroup of upper triangular matrices and let

\[ \varphi_0={\rm diag}(1,\ldots,1, q,\ldots, q,\ldots, q^{d-1},\ldots, q^{d-1})\in B(C), \]

where each entry $q^i$ appears $r$ times. Then a given element $N=(n_{ij})_{ij}\in {\operatorname {Lie} B}$ satisfies $N\varphi _0=q\varphi _0 N$ if and only if

\[ n_{ij}=0\quad \text{for}\ j\notin \{ir+1,\ldots, (i+1)r\}. \]

Scaling $\varphi _0$ by multiplication with elements of the center $Z\cong \mathbb {G}_m$, we obtain a closed embedding $\mathbb {G}_m\times \prod _{i=1}^{d-1}\mathbb {A}^{r^2}\hookrightarrow X_B$. The $B$-orbit of this closed subscheme is irreducible and of dimension

\begin{align*} \dim \bigg(\mathbb{G}_m\times \prod_{i=1}^{d-1}\mathbb{A}^{r^2}\bigg)+ \dim B-{\rm Stab}_B(\varphi_0)&=1+\dim B+\bigg((d-1)r^2-d\frac{r(r+1)}{2}\bigg)\\ &=1+\dim B+\frac{r}{2}(dr-(2r+d)). \end{align*}

In particular, we find that $X_B$ has an irreducible component of dimension strictly larger than $\dim B$ if $dr\geq 2r+d$. On the other hand, $X_B$ always has an irreducible component of dimension $\dim B$, namely $B\times \{0\}\subset B\times \operatorname {Lie} B$.

Let $G$ be a reductive group and let $P$ be a parabolic subgroup. We will write $\tilde X_P$ for the scheme representing the sheafification of the functor

(2.1)\begin{equation} R\mapsto \left\{(\varphi,N,g)\in (G\times\mathfrak{g})(R)\times G(R)/P(R)\, \left|\,{\begin{array}{@{}c@{}} (\varphi,N)\in X_G(R) \ \text{and}\\ \varphi\in g^{-1}Pg,\\ N\in {\rm Ad}(g^{-1})(\operatorname{Lie} P) \end{array}}\right\}.\right. \end{equation}

This is a closed $G$-invariant subscheme of $X_G\times G/P$ (where $G$ acts on $G/P$ by left translation). Then $(\varphi,N)\mapsto (\varphi,N,1)$ induces a closed embedding $X_P\hookrightarrow \tilde X_P$ which descends to an isomorphism

(2.2)\begin{equation} [X_P/P]\xrightarrow{\cong} [\tilde X_P/G]. \end{equation}

Moreover, the canonical projection $\tilde X_P \rightarrow X_G$ is $G$-equivariant and the induced morphism $[\tilde X_P/G]\rightarrow [X_G/G]$ agrees under the isomorphism (2.2) with the morphism $[X_P/P]\rightarrow [X_G/G]$ induced by $P\hookrightarrow G$. The following lemma is a direct consequence of this discussion.

We continue to assume that $G$ is reductive. We say that a point $(\varphi,N)\in G\times \mathfrak {g}$ is regular, if there are only finitely many Borel subgroups $B'\subset G$ such that $\varphi \in B'$ and $N\in \operatorname {Lie} B'$, that is, if (for one fixed choice of a Borel $B$) the point $(\varphi,N)$ has only finitely many pre-images under

\[ \left\{(\varphi,N,gB)\in G\times \mathfrak{g} \times G/B\,\left|\,\begin{array}{@{}c@{}} \varphi \in g^{-1}Bg,\\ N\in {\rm Ad}(g^{-1})(\operatorname{Lie} B) \end{array}\right.\right\}\longrightarrow G\times \mathfrak{g}. \]

As this morphism is proper and the fiber dimension is upper semi-continuous on the source, the regular elements form a Zariski open subset $(G\times \mathfrak {g})^{\rm reg}\subset G\times \mathfrak {g}$. Similarly, we can define a Zariski open subset

\[ X_G^{\rm reg}=X_G\cap (G\times\mathfrak{g})^{\rm reg}\subset X_G. \]

If $P\subset G$ is a parabolic subgroup, we write

\[ (P\times \operatorname{Lie} P)^{\rm reg}=(G\times \operatorname{Lie} G)^{\rm reg}\cap(P\times \operatorname{Lie} P) \]

and $X_P^{\rm reg}=X_P\cap X_G^{\rm reg}$. Moreover, we write $\tilde X_P^{\rm reg}$ for the pre-image of $X_G^{\rm reg}$ under $\tilde X_P\rightarrow X_G$. Then $[X_P^{\rm reg}/P]=[\tilde X_P^{\rm reg}/G]$ as stacks and the morphism $\tilde X_P^{\rm reg}\rightarrow X_G^{\rm reg}$ is by construction a finite morphism. Moreover, if we write $M$ for the Levi quotient of $P$, it is a direct consequence of the definition that the canonical projection $[X_P/P]\rightarrow [X_M/M]$ restricts to $[X_P^{\rm reg}/P]\rightarrow [X_M^{\rm reg}/M]$.

If $G=\operatorname {GL}_n$ and $P\subset G$ is a parabolic subgroup, then $G/P$ can be identified with the variety of flags of type $P$. In particular, we can identify $\tilde X_P$ with the variety of triples $(\varphi,N,\mathcal {F})$ consisting of $(\varphi,N)\in X_G$ and a $(\varphi,N)$-stable flag of type $P$. From now on we will often use this identification.

Proof. (i) Let $U\subset P$ denote the unipotent radical of $P$ and fix a section $M\hookrightarrow P$ of the canonical projection. We write $\mathfrak {u}\subset \mathfrak {p}$ for the Lie algebras of $U$ and $P$ and $\mathfrak {m}$ for the Lie algebra of $M$. Then we obtain a commutative diagram

where the horizontal arrow $\psi$ is induced by multiplication and the other two morphisms are the canonical projections. Let $r=\dim M$ and $s=\dim U$. Let $I\subset \Gamma (P\times \operatorname {Lie} P,\mathcal {O}_{P\times \operatorname {Lie} P})$ be the ideal defining $X_P\hookrightarrow P\times \operatorname {Lie} P$, that is, the ideal generated by the entries of the matrix ${\rm Ad}(\varphi )(N)-q^{-1}N$, where $\varphi$ and $N$ are the universal elements over $P$ and $\operatorname {Lie} P$. Then we deduce from the diagram that $I$ can be generated by elements $f_1,\ldots, f_r,g_1\ldots,g_s$ such that

\[ f_1,\ldots,f_r\in \Gamma(M\times \mathfrak{m},\mathcal{O}_{M\times \mathfrak{m}})\subset \Gamma(P\times \operatorname{Lie} P, \mathcal{O}_{P\times \operatorname{Lie} P}), \]

where $(f_1,\ldots,f_r)$ is the ideal defining $X_M\subset M\times \mathfrak {m}$. It follows that the ideal $(g_1,\ldots, g_s)$ is the ideal defining $X_P$ as a closed subscheme of $\pi ^{-1}(X_M)\cong \mathbb {A}^{2s}_{X_M}$.

Let us now write $K(g_1,\ldots, g_s)$ for the Koszul complex defined by $g_1,\ldots, g_s$ on the open subscheme $\pi ^{-1}(X_M)^{\rm reg}=\pi ^{-1}(X_M)\cap (P\times \operatorname {Lie} P)^{\rm reg}$ of $\pi ^{-1}(X_M)$. This is a finite complex of flat $\mathcal {O}_{X_M}$-modules and we claim that it is a resolution of $\mathcal {O}_{X_P^{\rm reg}}$. Indeed, $g_1,\ldots,g_s$ cut out the closed subscheme $X_P^{\rm reg}\subset \pi ^{-1}(X_M)^{\rm reg}$ which is of codimension $s$ by Lemma 2.5. As $\pi ^{-1}(X_M)^{\rm reg}$ is Cohen–Macaulay (it is an open subscheme of an affine space over $X_M$ and $X_M$ is Cohen–Macaulay as a consequence of Proposition 2.1) it follows from [Reference Grothendieck and DieudonnéEGAIV.1, Corollaire 16.5.6] that $g_1\ldots,g_s$ is a regular sequence and hence the Koszul complex is a resolution of its zeroth cohomology which is $\mathcal {O}_{X_P^{\rm reg}}$.

(ii) The fact that the squares are fiber products follows from the fact that $P'$ is the pre-image of $P'_M$ under $P\rightarrow M$. We show that the square on the right is Tor-independent. As in (i) we have a Koszul complex $K(g_1,\ldots, g_s)$ on $\pi ^{-1}(X_M)^{\rm reg}$ which is a $\mathcal {O}_{X^{\rm reg}_M}$-flat resolution of $\mathcal {O}_{X_P^{\rm reg}}$. Consider the closed embedding

(2.4)\begin{equation} X_{P'}^{\rm reg}\hookrightarrow \pi^{-1}(X_{P'_M})\cap (P\times \mathfrak{p})^{\rm reg}. \end{equation}

As (2.3) is cartesian, the restrictions of $g_1,\ldots, g_s$ to $\pi ^{-1}(X_{P'_M})\cap (P\times \operatorname {Lie} P)^{\rm reg}$ generate the ideal defining the closed embedding (2.4), and it remains to show that the pullback of the Koszul complex $K(g_1,\ldots, g_s)$ along (2.4) is a resolution of its zeroth cohomology group; that is, we need to show that $g_1,\ldots, g_s$ is a regular sequence in

\[ \mathcal{O}_{\pi^{-1}(X^{\rm reg}_{P'_M})\cap (P\times \operatorname{Lie} P)^{\rm reg},x}\quad \text{for all}\ x\in X_{P'}^{\rm reg}\subset \pi^{-1}(X^{\rm reg}_{P'_M})\cap (P\times \operatorname{Lie} P)^{\rm reg}. \]

Now

\[ \pi^{-1}(X_{P'_M}^{\rm reg})\cap (P\times \operatorname{Lie} P)^{\rm reg}\subset \pi^{-1}(X_{P'_M}^{\rm reg})\cong \mathbb{A}^{2s}_{X_{P'_M}^{\rm reg}} \]

is an open subscheme and hence it is Cohen–Macaulay as $X_{P'_M}^{\rm reg}$ is Cohen–Macaulay by Lemma 2.5. The claim now follows again from [Reference Grothendieck and DieudonnéEGAIV.1, Corollaire 16.5.6] and the fact that $X_{P'}^{\rm reg}$ is equidimensional of dimension $\dim \pi ^{-1}(X_{P'_M}^{\rm reg})-s$.

We reformulate the first claim of the lemma in terms of stacks.

Corollary 2.11 Let $P'\subset P\subset G$ be parabolic subgroups of $G$ and let $M$ be the Levi quotient of $P$ and $P'_M$ the image of $P'$ in M. Then the diagram

of stacks is cartesian.

Proof. Note that $P\rightarrow M$ induces an isomorphism $P/P'\cong M/P'_M$. As in (2.1), we define a closed $M$-invariant subscheme $Y_{P'_M}\subset X_M\times M/P'_M$ as the scheme representing the sheafification of the functor

\[ R\mapsto \left\{(\varphi,N,g)\in X_M(R)\times M(R)/P'_M(R) \left|{\begin{array}{c@{}} \varphi\in g^{-1}P'_Mg\ \text{and}\\ N\in {\rm Ad}(g^{-1})(\operatorname{Lie} P'_M) \end{array}}\right\}\right.. \]

Then we have a canonical isomorphism $[X_{P'_M}/P'_M]\cong [Y_{P'_M}/M]$. Similarly, we define a $P$-invariant closed subscheme $Y_{P'}\subset X_{P'}\times P/P'$ such that $[X_{P'}/P']\cong [Y_{P'}/P]$. Then the diagram

is cartesian by the above lemma. Let $U\subset P$ denote the unipotent radical of $P$, then it follows that

is cartesian diagram of stacks with $M$-action and with $M$-equivariant morphisms. The claim follows by taking the quotient by the $M$-action everywhere.

The objects introduced above have variants in the world of derived (or dg) schemes (see, for example, [Reference Drinfeld and GaitsgoryDG13, 0.6.8] and the references cited therein). The category of dg-schemes over $C$ canonically contains the category of $C$-schemes as a subcategory. For any linear algebraic group $G$ we write $\gamma _G:G\times \operatorname {Lie} G\rightarrow \operatorname {Lie} G$ for the morphism $(\varphi,N)\mapsto {\rm Ad}(\varphi )(N)-q^{-1}N$. We denote by $\mathbf {X}_G$ the fiber product

in the category of dg-schemes. This yields a dg-scheme $\mathbf {X}_G$ with underlying classical scheme ${}^{\rm cl}\mathbf {X}_G=X_G$. If $G$ is reductive then Proposition 2.1 implies that $\mathbf {X}_G=X_G$. Similarly, if $P\subset G$ is a parabolic subgroup of a reductive group, we denote by $\mathbf {X}_P^{\rm reg}\subset \mathbf {X}_P$ the open sub-dg-scheme with underlying topological space $X_P^{\rm reg}$. Then Lemma 2.5 implies that $\mathbf {X}_P^{\rm reg}=X_P^{\rm reg}$ is a classical scheme.

For any linear algebraic group $G$ we write $[\mathbf {X}_G/G]$ for the stack quotient of $\mathbf {X}_G$ by the canonical action of $G$. This is an algebraic dg-stackFootnote ² in the sense of [Reference Drinfeld and GaitsgoryDG13, 1.1]. Similarly to the case of schemes, we can view any algebraic stack as an algebraic dg-stack. Moreover, recall that every dg-stack $S$ has an underlying classical stack ${}^{\rm cl}S$.

If $G$ is reductive and $P\subset G$ is a parabolic we also consider the stacks $[\mathbf {X}_G^{\rm reg}/G]$ and $[\mathbf {X}_P^{\rm reg}/P]$. Then

\begin{align*} [\mathbf{X}_G/G]&=[X_G/G],\\ [\mathbf{X}_G^{\rm reg}/G]&=[X_G^{\rm reg}/G], \\ [\mathbf{X}_P^{\rm reg}/P]&=[X_P^{\rm reg}/P]. \end{align*}

We recall that a morphism $\mathbf {Y}_1\rightarrow \mathbf {Y}_2$ of dg-stacks is schematic if for all affine dg-schemes $\mathbf {Z}$ and all morphisms $\mathbf {Z}\rightarrow \mathbf {Y}_2$ the fiber product $\mathbf {Z}\times _{\mathbf {Y}_2}\mathbf {Y}_1$ is a dg-scheme (see [Reference Drinfeld and GaitsgoryDG13, 1.1.2]). A morphism of dg-schemes is called proper if the induced morphism of the underlying classical schemes is proper, and a morphism of algebraic dg-stacks is proper if the morphism of underlying classical stacks is proper in the sense of [Reference Laumon and Moret-BaillyLM00, Definition 7.11]. Similarly to the non-derived results above, we obtain the following lemma.

Proof. (i) As in the non-derived case we can rewrite $[\mathbf {X}_P/P]$ as $[\tilde {\mathbf {X}}_P/G]$, where $\tilde {\mathbf {X}}_P\subset \mathbf {X}_G\times G/P$ is the closed $G$-invariant sub-dg-scheme obtained by making

\[ \mathbf{X}_P\subset \mathbf{X}_G\times \{P\}\subset \mathbf{X}_G\times G/P \]

invariant under the $G$-action. More precisely, we can describe this dg-scheme as follows. Let us write $Z\subset G\times \operatorname {Lie} G\times G/P$ for scheme representing (the sheafification of) the functor

\[ R\longmapsto\{(\varphi,N,gP)\in (G\times\operatorname{Lie} G\times G/P)(R)\mid \varphi\in g^{-1}Pg,\ N\in{\rm Ad}(g^{-1})\operatorname{Lie} P\} \]

on $C$-algebras. By definition there is an isomorphism of stacks

\[ [Z/G]\cong [P\times\operatorname{Lie} P/P]. \]

Similarly, we write $Z'\subset \operatorname {Lie} G\times G/P$ for the scheme parametrizing pairs $(N,gP)$ such that $N\in {\rm Ad}(g^{-1})\operatorname {Lie} P$. Then $Z'\rightarrow G/P$ is a vector bundle (with fibers isomorphic to $\operatorname {Lie} P$) and we can define $\tilde {\mathbf {X}}_P$ as the derived fiber product

where $G/P\rightarrow Z'$ is the zero section and the right vertical arrow is the morphism given by $(\varphi,N,gP)\mapsto ({\rm Ad}(\varphi )N-q^{-1}N,gP)$. Now $[\mathbf {X}_P/P]=[\tilde {\mathbf {X}}_P/G]\rightarrow [\mathbf {X}_G/G]$, and the projection

(2.5)\begin{equation} \tilde{\mathbf{X}}_P\longrightarrow X_G \end{equation}

is a $G$-equivariant model for the canonical projection $[\mathbf {X}_P/P]\rightarrow [X_G/G]$ induced by $P\subset G$. The claim follows from this together with the observation that ${}^{\rm cl}\tilde {\mathbf {X}}_P=\tilde X_P$ and ${}^{\rm cl}[\mathbf {X}_P/P]=[X_P/P]$.

(ii) This is a direct consequence of the definition of the fiber product in the category of dg-schemes and the fact that $P'$ is the pre-image of $P'_M$ under the (flat) morphism $P\rightarrow M$.

Remark 2.13 Let $B\subset G$ be a Borel subgroup and let $U\subset B$ denote the unipotent radical. Then $\mathbf {X}_B\subset B\times \operatorname {Lie} B$ is in fact a (derived) subscheme of $B\times \operatorname {Lie} U$. Indeed, computing ${\rm Ad}(\varphi )N-q^{-1}N$ for the universal pair $(\varphi,N)$ over $B\times \operatorname {Lie} B$ and restricting to the Lie algebra of (a choice of) a maximal torus in $B$, we find that $\mathbf {X}_B\subset B\times \operatorname {Lie} U$. Using this observation, we mention the following variant of the construction in the proof of (i) above. Letting $\mathcal {N}_G$ denote the nilpotent cone of $G$, we can define $Y=Z\cap G\times \mathcal {N}_G\times G/B$. In fact $[Y/G]\cong [B\times \operatorname {Lie} U/B]$ and if we write $\tilde {G}\rightarrow G$ (respectively, $\tilde {\mathcal {N}}_G\rightarrow \mathcal {N}_G$) for the Grothendieck (respectively, Springer) resolution, then

\begin{align*} Y=\tilde G\times_{G/B}\tilde{\mathcal{N}}_G. \end{align*}

In particular, $Y$ is smooth and affine of relative dimension $\dim G$ over $G/B$. Then we can define $\tilde {\mathbf {X}}_B$ as a closed sub-dg-scheme of $Y$ similarly to its definition as a closed sub-dg-scheme of $Z$ in the proposition.

2.2 Derived categories of (quasi-)coherent sheaves

Given a scheme or a stack $X$ (or a derived scheme or a derived stack), we write ${\bf D}_{\rm QCoh}(X)$ for the derived category of quasi-coherent sheaves on $X$ (see [Reference Drinfeld and GaitsgoryDG13, 1.2] where it is denoted by ${\rm QCoh}(X)$). We write ${\bf D}^b_{\rm Coh}(X)$ for the full subcategory of objects that only have cohomology in finitely many degrees, which, moreover, is coherent.

If $X$ is a noetherian scheme then ${\bf D}^b_{\rm Coh}(X)$ coincides with the full subcategory of the derived category ${\bf D}(\mathcal {O}_X\text {-mod})$ of $\mathcal {O}_X$-modules, consisting of those complexes that have coherent cohomology and whose cohomology is concentrated in finitely many degrees. Similarly, if $X$ is a (classical) algebraic stack, then ${\bf D}^b_{\rm Coh}(X)$, and more generally ${\bf D}^+_{\rm QCoh}(X)$, agreesFootnote ³ with the definition of the bounded derived category of coherent sheaves, and, respectively, with the bounded-below derived category of quasi-coherent sheaves as defined in [Reference Laumon and Moret-BaillyLM00].

Lemma 2.14 Let $G$ be a reductive group and let $P$ be a parabolic subgroup with Levi quotient $M$. Let

\[ \alpha:[\mathbf{X}_P/P]\rightarrow [\mathbf{X}_M/M]\quad \text{and}\quad \beta:[\mathbf{X}_P/P]\rightarrow [\mathbf{X}_G/G] \]

denote the canonical morphisms. Then the maps

\begin{align*} & L\alpha^\ast:{\bf D}_{\rm QCoh}([\mathbf{X}_M/M])\longrightarrow {\bf D}_{\rm QCoh}([\mathbf{X}_P/P]),\\ & R\beta_\ast:{\bf D}_{\rm QCoh}([\mathbf{X}_P/P])\longrightarrow{\bf D}_{\rm QCoh}([\mathbf{X}_G/G]) \end{align*}

preserve the subcategories ${\bf D}^+_{\rm QCoh}(-)$ and ${\bf D}^b_{\rm Coh}(-)$.

Proof. In the case of $R\beta _\ast$ the claim directly follows from the fact that $\beta$ is proper and schematic. We prove the claim for $L\alpha ^\ast$. As the properties of belonging to ${\bf D}^+_{\rm QCoh}(-)$ or ${\bf D}^b_{\rm Coh}(-)$ can be checked over the smooth cover $\mathbf {X}_P$ of $[\mathbf {X}_P/P]$, it is enough to show that pullback along the morphism

\[ \alpha':\mathbf{X}_P\longrightarrow \mathbf{X}_M \]

preserves ${\bf D}^+_{\rm QCoh}(-)$ and ${\bf D}^b_{\rm Coh}(-)$. Moreover, both properties may be checked after forgetting the $\mathcal {O}_{\mathbf {X}_P}$-module structure, and only remembering the $\mathcal {O}_{P\times \operatorname {Lie} P}$-module structure. As in the proof of Lemma 2.9, we find that $\mathcal {O}_{\mathbf {X}_P}$ can be represented by a finite complex $\mathcal {F}_P^\bullet$ of flat $\mathcal {O}_{\mathbf {X}_M}$-modules that are coherent as $\mathcal {O}_{P\times \operatorname {Lie} P}$-modules, and $L\alpha '^\ast$ is identified with the functor $-\otimes ^{L}_{\mathcal {O}_{\mathbf {X}_M}}\mathcal {F}_P^\bullet$. The claim follows from this.

Let $P_1\subset P_2$ be parabolic subgroups of a reductive group $G$ with Levi quotients $M_i$, $i=1,2$. We write $P_{12}\subset M_2$ for the image of $P_1$ in $M_2$. Then $P_{12}\subset M_2$ is a parabolic subgroup with Levi quotient $M_1$. We obtain a diagram

(2.6)

where the upper left diagram is cartesian (in the category of dg-stacks).

Lemma 2.16 In the above situation we have a natural isomorphism

(2.7)\begin{equation} (R\beta_{2\,\ast}\circ L\alpha_{2}^\ast) \circ (R\beta_{12\,\ast}\circ L\alpha_{12}^\ast) \xrightarrow\cong R\beta_{1\,\ast}\circ L\alpha_{1}^\ast \end{equation}

of functors

\[ {\bf D}_{\rm QCoh}([\mathbf{X}_{M_1}/M_1])\longrightarrow {\bf D}_{\rm QCoh}([\mathbf{X}_{G}/G]). \]

Proof. The upper right square is (derived) cartesian and $\beta _{12}$ is schematic and proper; in particular, it is quasi-compact and quasi-separated. Hence, [Reference Drinfeld and GaitsgoryDG13, Propositions 1.3.6 and 1.3.10] implies that the natural base change morphism

\[ L\alpha_{2}^\ast \circ R\beta_{12\,\ast}\longrightarrow R\beta_\ast \circ L\alpha^\ast \]

is an isomorphism. We obtain the natural isomorphism

\begin{align*} & (R\beta_{2\,\ast}\circ L\alpha_{2}^\ast) \circ (R\beta_{12\,\ast}\circ L\alpha_{12}^\ast)=R\beta_{2\,\ast}\circ (L\alpha_{2}^\ast \circ R\beta_{12\,\ast})\circ L\alpha_{12}^\ast\\ &\quad \xrightarrow\cong R\beta_{2\,\ast}\circ (R\beta_\ast \circ L\alpha^\ast) \circ L\alpha_{12}^\ast =R\beta_{1\,\ast}\circ L\alpha_{1}^\ast. \end{align*}

We point out that, working only with classical schemes, we still obtain a natural transformation between the corresponding functors: if we consider the underlying classical stacks in diagram (2.6) and keep the same notation for the morphisms by abuse of notation, then there still is a natural base change morphism, but it is not necessarily an isomorphism as the fiber product might not be Tor-independent. However, it becomes an isomorphism when we restrict the functors to the regular locus, that is, we consider them as functors

\[ {\bf D}_{\rm QCoh}([X_{M_1}^{\rm reg}/M_1])\rightarrow {\bf D}_{\rm QCoh}([X_{G}^{\rm reg}/G]), \]

as in this case the classical and derived pictures coincide. Indeed, after the restriction to the regular locus the derived fiber product equals the classical fiber product by the Tor-independence in Lemma 2.9.

2.3 Duals

We continue to assume that $G$ is a reductive group over $C$ and fix a Borel subgroup $B\subset G$. Recall that we have defined a morphism

\[ \beta_B:[\mathbf{X}_B/B]\longrightarrow [X_G/G] \]

of (derived) stacks. In this subsection we will analyze the (derived) direct image of the structure sheaf under this morphism, which will play an important roleFootnote ⁴ later on. For a (derived) scheme or a (derived) stack $X$ we will write $\omega _X\in {\bf D}_{\rm QCoh}(X)$ for the dualizing complex and $\mathbb {D}_X(-)=R\mathcal {H}om_{\mathcal {O}_X}(-,\omega _X)$ for the usual duality functor. The main point of this section is to give evidence to the following conjecture (compare also [Reference Ben-Zvi, Chen, Helm and NadlerBCHN20, Conjecture 3.35] and [Reference ZhuZhu20, Conjecture 4.5.1 and Remark 4.5.4]).

Conjecture 2.17 The derived direct image $R\beta _\ast (\mathcal {O}_{[\mathbf {X}_B/B]})\in {\bf D}_{\rm Coh}^b([X_G/G])$ is concentrated in degree zero and there is an isomorphism

\[ R\beta_\ast(\mathcal{O}_{[\mathbf{X}_B/B]})\cong \mathbb{D}_{[X_G/G]} (R\beta_\ast(\mathcal{O}_{[\mathbf{X}_B/B]})). \]

In particular, the pullback of $R\beta _\ast (\mathcal {O}_{[\mathbf {X}_B/B]})$ to $X_G$ is concentrated in degree zero and is a maximal Cohen–Macaulay module.

Let us write $\tilde \beta _B:\tilde {\mathbf {X}}_B\rightarrow X_G$ for the $G$-equivariant model of $\beta _B$, as in (2.5). The conjecture is obviously equivalent to the claim that $R\tilde \beta _{B,\ast }\mathcal {O}_{\tilde {\mathbf {X}}_B}$ is concentrated in degree zero and that there is a $G$-equivariant isomorphism

\[ R\tilde\beta_{B,\ast}\mathcal{O}_{\tilde{\mathbf{X}}_B}\cong \mathbb{D}_{X_G}(R\tilde\beta_{B,\ast}\mathcal{O}_{\tilde{\mathbf{X}}_B})[-\dim G]. \]

This isomorphism then implies that the dual of the sheaf $R\tilde \beta _{B,\ast }\mathcal {O}_{\tilde {\mathbf {X}}_B}$ is concentrated in a single degree and hence, by [Sta, Tag 0B5A], it is a Cohen–Macaulay module. As its support obviously is all of $X_G$ it is then a maximal Cohen–Macaulay module.

As in Remark 2.13, we write $Y\rightarrow G\times \operatorname {Lie} G$ for the scheme parametrizing triples $(\varphi,N,gB)\in G\times \mathcal {N}_G\times G/B$ such that $\varphi \in g^{-1}Bg$ and $N\in {\rm Ad}(g^{-1})\operatorname {Lie} U$, where $U\subset B$ is the unipotent radical. Recall that $Y$ is smooth (of dimension $\dim G+\dim G/B$) as the projection to the flag variety $\operatorname {pr}:Y\rightarrow G/B$ is the composition of a $B$-bundle and a geometric vector bundle (or rank $\dim U$). We consider the diagram

and write $\mathcal {F}^\bullet =R\iota _{B,\ast }\mathcal {O}_{\tilde {\mathbf {X}}_B}$. The main observation is that $\mathcal {F}^\bullet$ is represented by a Koszul complex and hence is self-dual (up to a shift). In order to make this precise, recall that an algebraic representation of $B$ defines a $G$-equivariant vector bundle on $G/B$. We write $\mathcal {U}^\vee$ for the $G$-equivariant vector bundle on $G/B$ defined by the canonical $B$ representation on $\mathfrak {u}^\vee$. Here $\mathfrak {u}$ denotes the Lie algebra of $U$ (considered as a $C$-vector space), and $\mathfrak {u}^\vee$ denotes its dual. In particular, $\mathcal {U}^\vee$ admits a filtration whose graded pieces are the line bundles $\mathcal {L}_\alpha$ on $G/B$ associated to the negative (with respect to $B$) roots $\alpha$ of $G$.

Proof. (i) By definition the structure sheaf of the derived scheme $\mathbf {X}_B\subset B\times \operatorname {Lie} U$ is (as a complex of $\mathcal {O}_{B\times \operatorname {Lie} U}$-modules) quasi-isomorphic to the $B$-equivariant Koszul complex

\[ \cdots\longrightarrow \bigwedge\nolimits^i \mathfrak{u}^\vee\otimes_C \mathcal{O}_{B\times \operatorname{Lie} U} \longrightarrow \bigwedge\nolimits^{i-1}\mathfrak{u}^\vee\otimes \mathcal{O}_{B\times \operatorname{Lie} U}\longrightarrow\cdots \]

defined by the entries of ${\rm Ad}(\varphi )N-q^{-1}N$ for the universal pair $(\varphi,N)$ over $B\times \operatorname {Lie} U$ (where, by abuse of notation, we also write $\mathfrak {u}$ for the $C$-vector space underlying the scheme $\operatorname {Lie} U$). The claim follows from the identification $[Y/G]=[B\times \operatorname {Lie} U]$ (compare also Remark 2.13).

(ii) The claim on duality follows from the usual self-duality of Koszul complexes which is induced by the perfect pairing

\[ \bigwedge\nolimits^i \operatorname{pr}^\ast(\mathcal{U}^\vee)\times \bigwedge\nolimits^{\dim G/B-i} \operatorname{pr}^\ast(\mathcal{U}^\vee)\longrightarrow \bigwedge\nolimits^{\dim G/B} \operatorname{pr}^\ast(\mathcal{U}^\vee) \]

and the identification

\[ \bigwedge\nolimits^{\dim G/B} \operatorname{pr}^\ast{\mathcal{U}^\vee}= \operatorname{pr}^\ast\bigg(\bigwedge\nolimits^{\dim G/B}\mathcal{U}^\vee\bigg)= \operatorname{pr}^\ast \omega_{G/B}[-\dim G/B]=\omega_Y[-\dim Y].\]

Proof. The morphisms $\iota _G$ and $f$ satisfy Grothendieck duality, as $\iota _G$ is a complete intersection and $f$ factors into a complete intersection and a smooth morphism. Hence, we have the following isomorphisms in ${\bf D}_{\rm QCoh}(G\times \mathfrak {g})$:

\begin{align*} R\iota_{G,\ast}(R\tilde{\beta}_{B,\ast}(\mathcal{O}_{\tilde{\mathbf{X}}_B}))=Rf_\ast\mathcal{F}^\bullet &\cong Rf_\ast(\mathbb{D}_{Y}(\mathcal{F}^\bullet)[-\dim G])=Rf_\ast(\mathbb{D}_{Y} (\mathcal{F}^\bullet))[-\dim G]\\ &\cong \mathbb{D}_{G\times\mathfrak{g}}(Rf_\ast\mathcal{F}^\bullet)[-\dim G]\\ &=\mathbb{D}_{G\times\mathfrak{g}}(R\iota_{G,\ast}R\tilde{\beta}_{B,\ast} (\mathcal{O}_{\tilde{\mathbf{X}}_B}))[-\dim G]\\ &\cong R\iota_{G,\ast}(\mathbb{D}_{X_G}(R\tilde{\beta}_{B,\ast} (\mathcal{O}_{\tilde{\mathbf{X}}_B}))[-\dim G]). \end{align*}

Assuming Conjecture 2.19, the complex $Rf_\ast \mathcal {F}^\bullet$ is concentrated in degrees $(-\infty,0]$, and hence so is $R\tilde {\beta }_{B,\ast }(\mathcal {O}_{\tilde {\mathbf {X}}_B})$, as $\iota _{G}$ is affine. It follows that $\mathbb {D}_{X_G}(R\tilde {\beta }_{B,\ast }(\mathcal {O}_{\tilde {\mathbf {X}}_B}))$ is concentrated in degrees $[-\dim G,+\infty )$ and hence

\[ R\iota_{G,\ast}(\mathbb{D}_{X_G}(R\tilde{\beta}_\ast(\mathcal{O}_{\tilde{\mathbf{X}}_B}))[-\dim G]) \]

is concentrated in degrees $[0,+\infty )$. Hence, $Rf_\ast \mathcal {F}^\bullet$ is concentrated in degree zero and, again using that $\iota _G$ is affine, the same holds true for $R\tilde {\beta }_{B,\ast }(\mathcal {O}_{\tilde {\mathbf {X}}_B})$.

But as $R\tilde {\beta }_{B,\ast }(\mathcal {O}_{\tilde {\mathbf {X}}_B})$ is concentrated in a single degree the isomorphism

\[ R\iota_{G,\ast}(R\tilde{\beta}_{B,\ast}(\mathcal{O}_{\tilde{\mathbf{X}}_B}))\cong R\iota_{G,\ast}(\mathbb{D}_{X_G}(R\tilde{\beta}_{B,\ast} (\mathcal{O}_{\tilde{\mathbf{X}}_B}))[-\dim G]) \]

is in fact a $G$-equivariant isomorphism of sheaves on $G\times \mathfrak {g}$ (not just an isomorphism in the derived category) and hence restricts to a $G$-invariant isomorphism of sheaves

\[ R\tilde{\beta}_{B,\ast}(\mathcal{O}_{\tilde{\mathbf{X}}_B})\cong \mathbb{D}_{X_G}(R\tilde{\beta}_{B,\ast}(\mathcal{O}_{\tilde{\mathbf{X}}_B}))[-\dim G]. \]

on $X_G$. As we have seen above, these claims are equivalent to Conjecture 2.17.

Unfortunately we are not able to prove Conjecture 2.19 in general. We will instead give some evidence by proving the conjecture for ${\rm GL}_2$ and ${\rm GL}_3$.

As $G\times \mathfrak {g}$ is affine it is enough to prove that $H^i(G\times \mathfrak {g},Rf_\ast \mathcal {F}^\bullet )$ vanishes for $i>0$. This cohomology is computed by a spectral sequence

\[ E_1^{i,j}=H^j(Y,\mathcal{F}^i)\Longrightarrow H^{i+j}(G\times\mathfrak{g},Rf_\ast\mathcal{F}^\bullet), \]

and hence it is enough to show that $H^j(Y,\bigwedge \nolimits ^i\operatorname {pr}^\ast (\mathcal {U}^\vee ))=0\ \text {for}\ j>i$ (note that $\mathcal {F}^i=\bigwedge ^{-i}\operatorname {pr}^\ast (\mathcal {U}^\vee )$).

Proposition 2.21 Let $G={\rm GL}_2$ or $G={\rm GL}_3$. Then

\[ H^j\bigg(Y,\bigwedge\nolimits^i\operatorname{pr}^\ast(\mathcal{U}^\vee)\bigg)=0\quad \text{for}\ j>i. \]

In particular, Conjecture 2.17 holds true for ${\rm GL}_2$ and ${\rm GL}_3$.

Proof. We distinguish the cases $i>0$ and $i=0$. Let us prove the case $i>0$ first. We can compute this cohomology group after pushforward along the closed immersion $Y\hookrightarrow G\times \mathcal {N}_G\times G/B$ as follows. Let us write $\bar {\mathcal {U}}$ (respectively, $\bar {\mathcal {B}}$) for $G$-equivariant vector bundles on $G/B$ associated to the $B$-representations on $\mathfrak {g}/\mathfrak {b}$ (respectively, on $\mathfrak {g}/\mathfrak {u}$), where $\mathfrak {b}=\operatorname {Lie} B$. Note that $\mathcal {O}_Y$ is, as a sheaf on $G\times \mathcal {N}_G\times G/B$, quasi-isomorphic to the Koszul complex

\[ \cdots\longrightarrow \bigwedge\nolimits^i g^\ast(\bar{\mathcal{U}}^\vee \otimes\bar{\mathcal{B}}^\vee)\longrightarrow \bigwedge\nolimits^{i-1} g^\ast(\bar{\mathcal{U}}^\vee\otimes\bar{\mathcal{B}}^\vee)\longrightarrow \cdots \]

where again the $\bigwedge \nolimits ^i$ term appears in degree $-i$, and where $g:G\times \mathcal {N}_G\times G/B\rightarrow G/B$ denotes the canonical projection. By the projection formula (using that $g$ is affine and flat base change) we find that

\begin{align*} H^j(Y,\mathcal{F}^{-i})&=H^j\bigg(G\times\mathcal{N}_G\times G/B, \bigwedge\nolimits^ig^\ast(\mathcal{U})^\vee\otimes \mathcal{O}_Y\bigg)\\ &=H^j\bigg(G\times \mathcal{N}_G\times G/B, \bigwedge\nolimits^ig^\ast (\mathcal{U})^\vee\otimes \bigwedge\nolimits^\bullet g^\ast (\bar{\mathcal{U}}^\vee\otimes\bar{\mathcal{B}}^\vee)\bigg)\\ &=H^j\bigg(G/B,\bigwedge\nolimits^i\mathcal{U}^\vee\otimes \bigwedge\nolimits^\bullet (\bar{\mathcal{U}}^\vee\otimes \bar{\mathcal{B}}^\vee)\bigg)\otimes_C \Gamma(G\times \mathcal{N}_G,\mathcal{O}_{G\times\mathcal{N}}) \end{align*}

and again we can use a spectral sequence to compute this cohomology group. Hence, it is enough to show that

\[ H^\bullet\bigg(G/B,\bigwedge\nolimits^a\mathcal{U}^\vee\otimes \bigwedge\nolimits^b \bar{\mathcal{U}}^\vee\otimes \bigwedge\nolimits^c\bar{\mathcal{B}}^\vee\bigg)=0\quad \text{for } \bullet>a+b+c \text{ and } a\geq 1. \]

As $\bar {\mathcal {B}}^\vee$ is an extension of $\bar {\mathcal {U}}^\vee$ and a power of the trivial line bundle it is enough to prove that

\[ H^\bullet\bigg(G/B,\bigwedge\nolimits^a\mathcal{U}^\vee \otimes\bigwedge\nolimits^b \bar{\mathcal{U}}^\vee\otimes \bigwedge\nolimits^c\bar{\mathcal{U}}^\vee\bigg)=0\quad \text{for } \bullet>a+b+c \text{ and } a\geq 1. \]

We compute this cohomology group in terms of the cohomology of equivariant line bundles: note that the $G$-equivariant vector bundle $\bigwedge \nolimits ^a\mathcal {U}^\vee \otimes \bigwedge \nolimits ^b \bar {\mathcal {U}}^\vee \otimes \bigwedge \nolimits ^c\bar {\mathcal {U}}^\vee$ has a filtration whose subquotients are $G$-equivariant lines bundles $\mathcal {L}_{\lambda }\otimes \mathcal {L}_\mu \otimes \mathcal {L}_\nu =\mathcal {L}_{\lambda +\mu +\nu }$, where $\lambda$ is a sum of $a$ pairwise distinct positive roots,Footnote ⁵ $\mu$ is a sum of $b$ pairwise distinct negative roots and $\nu$ is a sum of $c$ pairwise distinct negative roots. It hence suffices to show that $H^\bullet (G/B,\mathcal {L}_{\lambda +\mu +\nu })$ vanishes in degrees higher than $a+b+c$ in this case. By the Borel–Bott–Weil theorem the cohomology of these line bundles is computed as follows. We write $w\cdot \kappa =w(\kappa +\rho )-\rho$ for the dot action of the Weyl group $W$ on $X^\ast (T)$, where $\rho$ is the half sum of the positive roots. Then there is either an element $1\neq w\in W$ such that $w\cdot \kappa =\kappa$, or a unique $w\in W$ such that $w\cdot \kappa$ is dominant. In the first case $H^\bullet (G/B,\mathcal {L}_{\kappa })=0$ and in the second case $H^\bullet (G/B,\mathcal {L}_{\kappa })$ is an irreducible $G$-representation concentrated in degree $\ell (w)$, where, as usual, $\ell (w)$ denotes the length of the Weyl group element $w$. We hence obtain a combinatorial claim about roots which allows us to check the vanishing of $H^j(Y,\bigwedge \nolimits ^i\operatorname {pr}^\ast (\mathcal {U}^\vee ))$.

Recall that we are assuming $i>0$. The above discussion implies that it is enough to prove the following claim. Let $a\geq 1$ and $b,c\geq 0$ and let $\lambda$ be a sum of $a$ pairwise distinct positive roots, $\mu$ be a sum of $b$ pairwise distinct negative roots and $\nu$ be a sum of $c$ pairwise distinct negative roots. If there is some (necessarily unique) $w\in W$ such that $w\cdot (\lambda +\mu +\nu )$ is dominant, then $\ell (w)\leq a+b+c$.

In the ${\rm GL}_2$ case this computation is rather trivial. In the ${\rm GL}_3$ case $G/B$ has dimension three and the claim is trivially satisfied for $a+b+c\geq 3$. We are left to check the cases $a+b+c=1$ and $a+b+c=2$.

We write $\alpha$ and $\beta$ for the two simple positive roots, and $s_\alpha,s_\beta \in W$ for the corresponding reflections.

In the case $a+b+c=1$ the condition $a\geq 1$ implies $b=c=0$, and hence we need to check that $\mathcal {L}_\alpha,\mathcal {L}_\beta$ and $\mathcal {L}_{\alpha +\beta }$ have cohomology only in degree $\leq 1$. In the case $a+b+c=2$ we need to check the cases $a=b=1, c=0$ (the case $a=c=1, b=0$ of course being equivalent to this case) and $a=2, b=c=0$. We hence need to show that the line bundles

\[ \mathcal{L}_0,\mathcal{L}_{\alpha-\beta},\mathcal{L}_{-\beta}, \mathcal{L}_{\beta-\alpha},\mathcal{L}_{-\alpha},\mathcal{L}_\alpha, \mathcal{L}_\beta,\mathcal{L}_{\alpha+\beta},\mathcal{L}_{\alpha+2\beta},\mathcal{L}_{2\alpha+\beta} \]

have cohomology only in degrees $\leq 2$.

More precisely, we have to check that for any of these weights $\kappa$, either $\kappa$ is fixed by some ${1\neq w}$ under the dot action, or $w\cdot \kappa$ is dominant for some $w$ of length less than or equal to $2$. This is done Table 1.

Table 1. The dot-action on sums of roots, I.

We are left to prove the proposition in the case $i=0$, that is, we need to show that $H^\bullet (Y,\mathcal {O}_Y)$ vanishes in positive degrees. Using the same criterion as above, we calculate $\mathcal {O}_Y$ as a Koszul complex which reduces us to proving that

\[ H^\bullet\bigg(G/B,\bigwedge\nolimits^b \bar{\mathcal{U}}^\vee\otimes \bigwedge\nolimits^c\bar{\mathcal{U}}^\vee\bigg)=0\quad \text{for}\ \bullet>b+c. \]

In the case $G={\rm GL}_2$ this is again trivially satisfied. In the case $G={\rm GL}_3$ this means we need to prove that $H^\bullet (G/B,\bar {\mathcal {U}}^\vee )$ vanishes in degrees higher than $1$ and $H^\bullet (G/B,\bar {\mathcal {U}}^\vee \otimes \bar {\mathcal {U}}^\vee )$ vanishes in degrees $>2$. We hence need to compute the cohomology of the lines bundles

\[ \mathcal{L}_{-\alpha},\mathcal{L}_{-\beta},\mathcal{L}_{-\alpha-\beta},\mathcal{L}_{-2\alpha}, \mathcal{L}_{-2\beta},\mathcal{L}_{-2\alpha-\beta},\mathcal{L}_{-\alpha-2\beta},\mathcal{L}_{-2\alpha-2\beta}. \]

Using Borel–Bott–Weil again, these cohomologies are computed using Table 2, where $w_0=s_\alpha s_\beta s_\alpha$ is the longest element. It is a direct consequence that the cohomology $H^\bullet (G/B,\bar {\mathcal {U}}^\vee )$ is concentrated in degrees less than or equal to $1$, whereas the cohomology $H^\bullet (G/B,\bar {\mathcal {U}}^\vee \otimes \bar {\mathcal {U}}^\vee )$ is concentrated in degrees less than or equal to $2$ if and only if the boundary map

\[ H^2(G/B,\mathcal{L}_{-2\alpha-\beta}\oplus\mathcal{L}_{-\alpha-2\beta}) \longrightarrow H^3(G/B,\mathcal{L}_{-2\alpha-2\beta}) \]

is surjective (or equivalently non-zero as the target of the map is an irreducible $G$-representation). By Serre duality this is equivalent to the injectivity of the induced map

(2.8)\begin{equation} H^0(G/B,\mathcal{L}_{0})\longrightarrow H^1(G/B,\mathcal{L}_{-\alpha}\oplus\mathcal{L}_{-\beta}). \end{equation}

But this map is the boundary map in cohomology induced by a non-split extension

(2.9)\begin{equation} 0\longrightarrow \mathcal{L}_{-\alpha}\oplus\mathcal{L}_{-\beta}\longrightarrow \mathcal{E}\longrightarrow \mathcal{L}_{0}\longrightarrow 0. \end{equation}

The fact that this extension is non-split implies that

\[ H^0(G/B,\mathcal{E})\longrightarrow H^0(G/B, \mathcal{L}_{0}) \]

is the zero map (as $\mathcal {L}_0$ is the trivial line bundle the canonical map $\Gamma (G/B,\mathcal {L}_0)\otimes \mathcal {O}_{G/B}\rightarrow \mathcal {L}_0$ is an isomorphism, and if $H^0(G/B,\mathcal {E})\longrightarrow H^0(G/B, \mathcal {L}_{0})$ was non-zero it necessarily would be an isomorphism which would imply that the canonical map $\Gamma (G/B,\mathcal {E})\otimes \mathcal {O}_{G/B}\rightarrow \mathcal {E}$ would induce a splitting of (2.9)) which in turn implies that (2.8) is injective. It follows that $H^\bullet (Y,\mathcal {O}_Y)=0$ for $\bullet >0$ as claimed.

Table 2. The dot-action on sums of roots, II.

3.1 Categories of smooth representations

Let us write ${\rm Rep}(G)$ for the category of smooth representations of $G$ on $C$ vector spaces. It is well known that ${\rm Rep}(G)$ has a decomposition into Bernstein blocks

\[ {\rm Rep}(G)=\prod_ {[M,\sigma]\in\Omega(G)}{\rm Rep}_{[M,\sigma]}(G), \]

where $\Omega (G)$ is a set of equivalence classes of a Levi $M$ of $G$ and a cuspidal representation $\sigma$ of $M$ (see [Reference BernsteinBer92, III, 2.2], for example). We will restrict our attention to the Bernstein component ${\rm Rep}_{[T,1]}(G)$, where $1$ is the trivial representation of the torus $T$. Given $\pi \in {\rm Rep}(G)$ we write $\pi _{[T,1]}$ for its image under the projection to ${\rm Rep}_{[T,1]}(G)$. Moreover, we will write $\mathfrak {Z}_G$ for the center of the category ${\rm Rep}_{[T,1]}(G)$; then

\[ \mathfrak{Z}_G\xrightarrow{\cong}\Gamma(\check G/\hspace{-.1cm}/\check G, \mathcal{O}_{\check G/\hspace{-.1cm}/\check G}) \]

(see below for an explicit description). This isomorphism allows us to identify the category $\mathfrak {Z}_G\text {-mod}$ of $\mathfrak {Z}_G$-modules with the category ${\rm QCoh}(\check T/W)$ of quasi-coherent sheaves on the adjoint quotient $\check G/\hspace {-0.1cm}/\check G=\check T/W$ of $\check G$, and the category $\mathfrak {Z}_G\text {-mod}_{\rm fg}$ of finitely generated $\mathfrak {Z}_G$-modules with the category ${\rm Coh}(\check T/W)$ of coherent sheaves on $\check T/W$. We obtain an identification of derived categories

(3.2)\begin{equation} \begin{aligned} {\bf D}(\mathfrak{Z}_G\text{-mod}) & \cong {\bf D}_{\rm QCoh}(\check T/W),\\ {\bf D}^b(\mathfrak{Z}_G\text{-mod}_{\text{fg}}) & \cong {\bf D}^b_{\rm Coh}(\check T/W). \end{aligned} \end{equation}

We use these identifications and the morphism $\bar \chi :[X_{\check G}/\check G]\rightarrow \check T/W$ to make ${\bf D}^+_{\rm QCoh}([X_{\check G}/\check G])$ and ${\bf D}^b_{\rm Coh}([X_{\check G}/\check G])$ into $\mathfrak {Z}_G$-linear categories.

If $\mathbb {P}\subset \mathbb {G}$ is a parabolic subgroup with Levi quotient $\mathbb {M}$, we write

\[ \iota_P^G={\rm Ind}_P^G(\delta_P^{1/2}\otimes -):{\rm Rep}(M)\longrightarrow{\rm Rep}(G) \]

for the normalized parabolic induction, and $\iota _{\overline {P}}^G$ for normalized parabolic induction of the opposite parabolic $\overline {P}$ of $P$ (note that the normalization uses the choice of $q^{1/2}$). These functors are exact and restrict to functors

\[ {\rm Rep}_{[T_M,1]}(M)\longrightarrow {\rm Rep}_{[T,1]}(G) \]

(for any choice of a maximal split torus $T_M$ of $M$). Using a splitting $\mathbb {M}\hookrightarrow \mathbb {P}\subset \mathbb {G}$ to the projection, we obtain a morphism

\[ \check M/\hspace{-.1cm}/\check M\longrightarrow \check G/\hspace{-.1cm}/\check G \]

which is obviously independent of the choice of $\mathbb {M}\hookrightarrow \mathbb {G}$. Then the functors $\iota _P^G$ and $\iota _{\overline {P}}^G$ are linear with respect to the morphism

\[ \mathfrak{Z}_G\cong\Gamma(\check G/\hspace{-.1cm}/\check G, \mathcal{O}_{\check G/\hspace{-.1cm}/\check G})\longrightarrow \Gamma(\check M/\hspace{-.1cm}/\check M,\mathcal{O}_{\check M/\hspace{-.1cm}/\check M})\cong\mathfrak{Z}_M \]

(see below for details).

Let us write ${\bf D}({\rm Rep}_{[T,1]}(G))$ (respectively, ${\bf D}^+({\rm Rep}_{[T,1]}(G))$) for the derived category (respectively, for the bounded-below derived category) of ${\rm Rep}_{[T,1]}(G)$. Moreover, we write ${\bf D}^b({\rm Rep}_{[T,1],{\rm fg}}(G))$ for the full subcategory of complexes whose cohomology is concentrated in bounded degrees and is finitely generated as a $C[G]$-module. Then $\iota _P^G$ (respectively, $\iota _{\overline {P}}^G$) induce functors

\begin{align*} {\bf D}({\rm Rep}_{[T_M,1]}(M))&\longrightarrow {\bf D}({\rm Rep}_{[T,1]}(G)),\\ {\bf D}^+({\rm Rep}_{[T_M,1]}(M))&\longrightarrow {\bf D}^+({\rm Rep}_{[T,1]}(G)),\\ {\bf D}^b({\rm Rep}_{[T_M,1],{\rm fg}}(M))&\longrightarrow {\bf D}^b({\rm Rep}_{[T,1],{\rm fg}}(G)), \end{align*}

which we will also denote by $\iota _P^G$ (respectively, $\iota _{\overline {P}}^G$).

Given two parabolic subgroups $\mathbb {P}_1\subset \mathbb {P}_2$ of $\mathbb {G}$ with Levi quotients $\mathbb {M}_1$ and $\mathbb {M}_2$, We write $\mathbb {P}_{12}$ for the image of $\mathbb {P}_1$ in $\mathbb {M}_2$. Then we have natural isomorphisms

(3.3)\begin{equation} \begin{aligned} \iota_{P_{2}}^{G}\circ \iota_{P_{12}}^{M_2} & \longrightarrow \iota_{P_{1}}^{G},\\ \iota_{\overline{P}_{2}}^{G}\circ \iota_{\overline{P}_{12}}^{M_2} & \longrightarrow \iota_{\overline{P}_{1}}^{G} \end{aligned} \end{equation}

of functors ${\bf D}({\rm Rep}_{[T_{M_1},1]}(M_1))\rightarrow {\bf D}({\rm Rep}_{[T,1]}(G))$.

Finally, recall that a Whittaker datum is a $G$-conjugacy class of tuples $(\mathbb {B},\psi )$, where $\mathbb {B}\subset \mathbb {G}$ is a Borel subgroup and $\psi :N\rightarrow C^\times$ is a generic character of $N=\mathbb {N}(F)$, where $\mathbb {N}\subset \mathbb {B}$ is the unipotent radical. As above, we fix the choice of a Borel subgroup $\mathbb {B}$ and a maximal split torus $\mathbb {T}\subset \mathbb {G}$. For a parabolic $\mathbb {P}\subset \mathbb {G}$ containing $\mathbb {B}$ with Levi quotient $\mathbb {M}$, we write $\psi _M:N_M\rightarrow C^\times$ for the restriction of $\psi$ to the unipotent radical $N_M\subset N$ of the Borel $B_M=B\cap M$ of $M$. Note that the $M$-conjugacy class of $(B_M,\psi _M)$ does not depend on the choice of $\mathbb {M}\hookrightarrow \mathbb {G}$ (i.e. on the choice of $\mathbb {T}$).

We can describe the above categories of representations in terms of modules over Iwahori–Hecke algebras. In order to do so, let us fix a hyperspecial vertex in the apartment of the Bruhat–Tits building of $G$ defined by the maximal torus $\mathbb {T}$, that is, we fix $\mathcal {O}_F$-models of $(\mathbb {G},\mathbb {T})$. The choice of a Borel $\mathbb {B}$ then defines an Iwahori subgroup $I\subset G$. We write ${\rm Rep}^I G$ for the category of smooth representations $\pi$ of $G$ on $C$-vector spaces that are generated by their Iwahori fixed vectors $\pi ^I$ and ${\rm Rep}_{\rm fg}^I G\subset {\rm Rep}^I G$ for the full subcategory of representations that are finitely generated (as $C[G]$-modules). It is well known that the category ${\rm Rep}^I G$ does not depend on the choice of $I$ and agrees with the Bernstein block ${\rm Rep}_{[T,1]}(G)$.

Let $\mathcal {H}_G=\mathcal {H}(G,I)={\rm End}_G(\text {c-ind}_I^G\mathbf {1}_I)$ denote the Iwahori–Hecke algebra. Then

\[ \pi\longmapsto \pi^I={\rm Hom}_G(\text{c-ind}_I^G\mathbf{1}_I,\pi) \]

induces an equivalence of categories between ${\rm Rep}_{[T,1]}(G)={\rm Rep}^I G$ and the category $\mathcal {H}_G\text {-mod}$ of $\mathcal {H}_G$-modules. This equivalence identifies ${\rm Rep}^I_{\rm fg} G$ and the full subcategory $\mathcal {H}_G\text {-mod}_{\text {fg}}\subset \mathcal {H}_G\text {-mod}$ of finitely generated $\mathcal {H}_G$-modules. Moreover, it identifies the center $\mathfrak {Z}_G$ of ${\rm Rep}_{[T,1]}(G)$ with the center of the Iwahori–Hecke algebra $\mathcal {H}_G$. Then we have an isomorphism

\[ \mathfrak{Z}_G\cong C[X_\ast(\mathbb{T})]^W=C[X^\ast(\check T)]^W= \Gamma(\check T/W,\mathcal{O}_{\check T/W})=\Gamma(\check G/\hspace{-0.1cm}/\check G, \mathcal{O}_{\check G/\hspace{-0.1cm}/\check G}) \]

(see, for example, [Reference Haines, Kottwitz and PrasadHKP10, Lemma 2.3.1]), which is in fact independent of the choice of the Iwahori $I$.

Given a representation $\pi \in {\rm Rep}^IG$ and a $\mathfrak {Z}_G$-module $\rho$, we will sometimes (by abuse of notation) write $\pi \otimes _{\mathfrak {Z}_G}\rho$ for the pre-image of the $\mathcal {H}_G$-module $\pi ^I\otimes _{\mathfrak {Z}_G}\rho$ under the equivalence of categories ${\rm Rep}^I G\cong \mathcal {H}_G\text {-mod}$ (and similarly for corresponding derived functors).

Remark 3.1 Note that if $\mathbb {G}=\mathbb {T}$ is a split torus, then $I=I_T=T^\circ$ is the unique maximal compact subgroup of $T$ and we have canonical identifications

(3.4)\begin{equation} C[X_\ast(\mathbb{T})]\cong C[T/T^\circ]=\mathcal{H}_T, \end{equation}

where the first isomorphism is given by $\mu \mapsto \mu (\varpi )$ for the choice of a uniformizer $\varpi$ of $F$ (note that this isomorphism is independent of this choice). We often use this isomorphism to identify unramified characters and $\mathcal {H}_T$-modules.

Let $\mathbb {P}\subset \mathbb {G}$ be a parabolic subgroup containing $\mathbb {B}$ with Levi quotient $\mathbb {M}$ and write $P=\mathbb {P}(F)$ and $M=\mathbb {M}(F)$. Set $I_M=I_G\cap M$, which is an Iwahori subgroup of $M$; in particular, we have ${\rm Rep}_{[T_M,1]}(M)={\rm Rep}^{I_M}M$. There is a canonical embedding $\mathcal {H}_M\hookrightarrow \mathcal {H}_G$ such that the diagrams

(3.5)

commute. Note that this is equivalent to the commutativity of the diagram

(3.6)

Here $r^G_P(-)$ is the normalized Jacquet module which is the left adjoint functor to $\iota _P^G(-)$. It is also the right adjoint functor to $\iota _{\overline {P}}^G(-)$ by Bernstein's second adjointness theorem. By abuse of notation we will often write $\iota _{P}^G$ (respectively, $\iota _{\overline {P}}^G$) for the functors ${\rm Hom}_{\mathcal {H}_M}(\mathcal {H}_G,-)$ (respectively, $\mathcal {H}_G\otimes _{\mathcal {H}_M}-$) on Hecke modules.

The embedding $\mathcal {H}_M\subset \mathcal {H}_G$ induces an embedding $\mathfrak {Z}_G\subset \mathfrak {Z}_M$, where $\mathfrak {Z}_M$ is the center of ${\rm Rep}_{[T_M,1]}(M)$ which is identified with the center of $\mathcal {H}_M$, such that the canonical diagram

commutes. We deduce that $\iota _P^G$ and $\iota _{\overline {P}}^G$ are $\mathfrak {Z}_G$-linear. In particular, for a $\mathfrak {Z}_G$-module $\rho$ we obtain natural isomorphisms

(3.7)\begin{equation} \begin{aligned} \iota_P^G(-\otimes_{\mathfrak{Z}_G}\rho) & \longrightarrow \iota_P^G(-)\otimes_{\mathfrak{Z}_G}\rho,\\ \iota_{\overline{P}}^G(-\otimes_{\mathfrak{Z}_G}\rho) & \longrightarrow \iota_{\overline{P}}^G(-) \otimes_{\mathfrak{Z}_G}\rho, \end{aligned} \end{equation}

and similarly for the corresponding functors on the derived category.

3.2 The main conjecture

Using the notation introduced above, we state the following conjecture. Variants of the conjecture have been around in representation theory in recent years. A proof of the conjecture is announced in the work of Ben-Zvi et al. [Reference Ben-Zvi, Chen, Helm and NadlerBCHN20] and Zhu [Reference ZhuZhu20].

Let us point out that the $\mathfrak {Z}_G$-linearity of the conjectured functor $R_G^\psi$ implies that for each $\rho \in {\bf D}^+(\mathfrak {Z}_G\text {-}{\rm mod})$ there is a natural isomorphism

\[ \psi_{G,\rho}: R_G^\psi(-\otimes_{\mathfrak{Z}_G}^L\rho)\xrightarrow{\cong} R_G^\psi(-)\otimes^L_{\mathcal{O}_{[X_{\check G}/\check G]}}L\bar\chi_{G}^\ast\rho \]

of functors ${\bf D}^+({\rm Rep}_{[T,1]}(G))\rightarrow {\bf D}^+_{\rm QCoh}([X_{\check G}/\check G])$ which is functorial in $\rho$ (in the obvious sense). Moreover, given $\mathbb {P}\subset \mathbb {G}$ as in (ii), the $\mathfrak {Z}_{G}$-linearity of the natural isomorphism $\xi _{P}^{G}$ implies that the natural transformations $\psi _{M,\rho }$ and $\psi _{G,\rho }$ are compatible with parabolic induction. We do not spell this out explicitly in terms of commutative diagrams.

Remark 3.3 (a) We expect that the conjectured functor $R_G^\psi$ induces a functor

\[ {\bf D}^b({\rm Rep}_{[T,1],{\rm fg}}(G))\longrightarrow {\bf D}^b_{\rm Coh}([X_{\check G}/\check G]). \]

This would allow us to extend the functor to the full derived category ${\bf D}({\rm Rep}_{[T,1]}(G))$: as we have an equivalence of categories ${\rm Rep}_{[T,1],{\rm fg}}(G)\cong \mathcal {H}_G\text {-mod}_{\text {fg}}$ and as $\mathcal {H}_G$ has finite global dimension (see [Reference BernsteinBer92, 4. Theorem 29]), the full derived category ${\bf D}({\rm Rep}_{[T,1]}(G))$ is the ind-completion of ${\bf D}^b({\rm Rep}_{[T,1],{\rm fg}}(G))$. Hence, the conjectured functor would extend to a fully faithful and exact functor

\[ {\bf D}({\rm Rep}_{[T,1]}(G))\longrightarrow {\rm IndCoh}([X_{\check G}/\check G]), \]

where ${\rm IndCoh}([X_{\check G}/\check G])$ is the ind-completion of ${\bf D}^b_{\rm Coh}([X_{\check G}/\check G])$. Note that this category differs from ${\bf D}_{\rm QCoh}([X_{\check G}/\check G])$, as $X_{\check G}$ is singular. However, there is a canonical equivalence

\[ {\rm IndCoh}^+([X_{\check G}/\check G])\xrightarrow{\cong}{\bf D}^+_{\rm QCoh}([X_{\check G}/\check G]) \]

(see, for example, [Reference Drinfeld and GaitsgoryDG13, 3.2.4]). In particular, restricting to bounded-below objects, the conjecture that the (as yet hypothetical) functor $R_G^\psi$ is fully faithful does not depend on whether we consider it as a functor with values in ${\rm IndCoh}^+([X_{\check G}/\check G])$ or with values in ${\bf D}^+_{\rm QCoh}([X_{\check G}/\check G])$. We hence arrive at a conjecture that parallels the formulation of the geometric Langlands program (see [Reference GaitsgoryGai15]). Also the conjectured compatibility with parabolic induction agrees with the compatibility with parabolic induction in [Reference GaitsgoryGai15]. See also the formulation given in [Reference ZhuZhu20, Conjecture 4.4.5].

(b) Recall that an L-parameter for $G$ that is trivial on inertia is a $\check G$-conjugacy class $[\varphi,N]$ of $(\varphi,N)\in X_{\check G}(C)$ with $\varphi$ semi-simple. We write $S_{[\varphi,N]}=C_{[\varphi,N]}/C^\circ _{[\varphi,N]}$ for the quotient of the centralizer of $(\varphi,N)$ by its connected component of the identity. By the classification of Kazhdan and Lusztig [Reference Kazhdan and LusztigKL87, Theorem 7.12] the irreducible representations in ${\rm Rep}^IG$ (and the simple objects in $\mathcal {H}_G\text {-mod}$) are in bijection with pairs $([\varphi,N],\rho )$, where $[\varphi,N]$ is an L-parameter and $\rho$ runs through a certain set of irreducible representation of $S_{[\varphi,N]}$. This parametrization depends on an additional choice that corresponds to the choice of a Whittaker datum $(\mathbb {B},\psi )$. More precisely, the classification in [Reference Kazhdan and LusztigKL87] (which in the case of $\operatorname {GL}_n$ coincides with the Bernstein–Zelevinsky classification [Reference Bernstein and ZelevinskyBZ77]) associates to $([\varphi,N],\rho )$ an indecomposable representation (respectively, Hecke module) $\pi ^\psi _{[\varphi,N],\rho }$ which has a unique irreducible quotient. Conjecture 3.2 should have the following relation with this classification. For simplicity we only treat the case of regular semi-simple $\varphi$; the general case seems to be much more involved (see remark (d) in § 4.7 for some discussion in the case of ${\rm GL}_n$)).

Given $[\varphi,N]$, let us write

\[ X_{\check G,[\varphi,N]}^\circ\subset X_{\check G} \]

for the $\check G$-orbit of $(\varphi,N)$. Moreover, we denote by

\[ X_{\check G,[\varphi,N]}=\overline{X_{\check G,[\varphi,N]}^\circ} \]

its Zariski closure. As we assume that $\varphi$ is regular semi-simple we can, given an irreducible representation $\rho$ of $S_{[\varphi,N]}$ on a finite-dimensional $C$-vector space, use $\rho$ to define a $\check G$-equivariant coherent sheaf

\[ \tilde{\mathcal{F}}_{[\varphi,N],\rho}\in{\rm Coh}(X_{\check G,[\varphi,N]}) \]

which hence defines a coherent sheaf $\mathcal {F}_{[\varphi,N],\rho }$ on the closed substack

\[ [X_{\check G,[\varphi,N]}/\check G]\subset [X_{\check G}/\check G]. \]

We then expect that the conjectured functor $R_G$ has the property

\[ R_G^\psi(\pi_{[\varphi,N],\rho}^\psi)=\mathcal{F}_{[\varphi,N],\rho}. \]

If the L-parameter $[\varphi,N]$ is generic, there is a unique $\psi$-generic representation $\pi$ in the L-packet defined by $[\varphi,N]$. With the above notation this representation is the representation

\[ \pi=\pi^\psi_{[\varphi,N],{\rm trivial}}. \]

Then the expected formula above specializes to

\[ R_G^\psi(\pi)=\mathcal{O}_{[X_{\check G,[\varphi,N]}/\check G]}. \]

(c) We point out that the conjectured functor $R_G^\psi$ will not be essentially surjective. In fact this is already obvious in the case where $G=T$ a split torus. Here $R_T=R_T^\psi$ is the derived version of the functor

\[ \mathcal{H}_T\text{-}{\rm mod}\cong{\rm QCoh}(\check T)\longrightarrow {\rm QCoh}([\check T/\check T]). \]

The morphism on the right-hand side is the embedding given by equipping a quasi-coherent sheaf with the trivial $\check T$-action. Obviously $\check T$-equivariant sheaves with non-trivial $\check T$-action are not in the essential image.

There is also a second obstruction for essential surjectivity for general $G$ (i.e. $G$ is not assumed to be a torus). Let $[\varphi,N]$ be an L-parameter such that $\varphi$ is semi-simple but not regular semi-simple, and assume $N=0$ for simplicity. Similarly to (b), we write $X_{\check G,[\varphi,0]}\subset X_{\check G}$ for the closed subset of $(\varphi ',0)$ such that $\varphi '^{\rm ss}$ is in the $\check G$-orbit of $\varphi ^{\rm ss}$. Then the structure sheaf of the closed substack

\[ [X_{\check G,[\varphi,0]}^{\rm ss}/\check G]\subset [X_{\check G,[\varphi,0]}/\check G] \]

of pairs $(\varphi ',0)$ where $\varphi '$ is (pointwise) semi-simple should not be in the essential image of the functor $R_G$. The same phenomenon should apply to all $(\varphi,N)$ with $\varphi$ not regular semi-simple (not only in the case $N=0$), but the definition of $X_{\check G,[\varphi,N]}$ is more involved in general (see remark (d) in § 4.7 for some details).

Following Fargues and Scholze [Reference Fargues and ScholzeFS21] and Zhu [Reference ZhuZhu20, 4.6], the failure of essential surjectivity should be fixed by replacing the category of smooth representations by a larger category.

(d) Finally, we point out that in the conjecture it is necessary to pass to derived categories. Heuristically this can be explained by the fact that flat morphisms on the representation theory side correspond to non-flat morphisms on the side of stacks: for example, $\mathcal {H}_G$ is flat over its center, whereas the canonical morphism

\[ \bar\chi_G:[X_{\check G}/\check G]\longrightarrow \check T/W \]

is not flat (as it maps some irreducible components to proper closed subschemes of $\check T/W$). Moreover, we will see below that in the case of ${\rm GL}_n(F)$ the trivial representation will be mapped to a complex concentrated in cohomological degree $1-n$ (see Remark 4.43). Hence, without passing to derived categories, the functor cannot be fully faithful. The canonical t-structures on the source (or target) should correspond to an exotic t-structure on the other side. However, we have no idea how this t-structure could be described intrinsically. Moreover, the formulation of the conjecture needs the passage to derived schemes or derived stacks: as parabolic induction is transitive (in the sense that (3.3) is an isomorphism), the base change morphism (2.7) has to be an isomorphism as well. However, in the world of classical schemes the corresponding cartesian diagram is not Tor-independent in general.

3.3 A generalization of the conjecture

Conjecture 3.2 in fact is a special case of a more general conjecture about the category ${\rm Rep}(G)$, instead of the Bernstein block ${\rm Rep}_{[T,1]}(G)$. Let us describe this generalization. A similar generalization is conjectured by Zhu [Reference ZhuZhu20, Conjecture 4.5.1]. The generalization stated here can also be viewed as a special case of the main conjecture [Reference Fargues and ScholzeFS21, Conjecture I.10.2] of Fargues and Scholze.

We continue to assume that $\mathbb {G}$ is a split reductive group with dual group $\check G$. Let us write $W_F$ for the Weil group of $F$ and $I_F\subset W_F$ for the inertia group. We define the space of $\check G$-valued Weil–Deligne representations to be the scheme $X_{\check G}^{\rm WD}$ representing the functor

\[ R\longmapsto\left\{\rho:W_F\rightarrow \check G(C),\ N\in \operatorname{Lie}\check G\,\left|\!\begin{array}{c@{}} \rho|_{J}\ \text{is trivial for some}\ J\subset I_F\ \text{open}\\ {\rm Ad}(\rho(\sigma))(N)=q^{-\|\sigma\|}N \end{array}\right.\right\} \]

on $C$-algebras $R$. Here $\|-\|:W_F\rightarrow \mathbb {Z}$ is the usual projection. It is easy to see that $X_{\check G}^{\rm WD}$ is an infinite disjoint union of affine schemes and is equipped with a $\check G$-action via conjugation on $\rho$ and via the adjoint action on $N$. The space of Weil–Deligne representations $X_{\check G}^{\rm WD}$ in fact agrees with the fiber over $C$ of the moduli space of L-parameters $\underline {Z}^1(W_F^\circ,\check G)$ studied in work of Dat et al. [Reference Dat, Helm, Kurinczuk and MossDHKM20] and is defined and studied as well in [Reference ZhuZhu20, 3.1].

Similarly, for every parabolic subgroup $\check P\subset \check G$ we can define the scheme $X_{\check P}^{\rm WD}$ and the derived scheme ${\bf X}_{\check P}^{\rm WD}$ that come equipped with $\check P$-actions.

The inclusion $\check P\hookrightarrow \check G$ and the projection $\check P\rightarrow \check M$ onto the Levi quotient $\check M$ of $\check P$ induce morphisms

(3.8)\begin{equation} \begin{aligned} \beta_{\check P}^{\rm WD}:[{\bf X}_{\check P}^{\rm WD}/\check P] & \longrightarrow [X_{\check G}^{\rm WD}/\check G],\\ \alpha_{\check P}^{\rm WD}:[{\bf X}_{\check P}^{\rm WD}/\check P] & \longrightarrow [X_{\check M}^{\rm WD}/\check M] \end{aligned} \end{equation}

of the respective stack quotients. Moreover, we will write $X_{\check G}^{\rm WD}/\hspace {-0.1cm}/\check G$ for the GIT quotient of $X_{\check G}^{\rm WD}$ by the $\check G$-action. As in the case of the space of $(\varphi,N)$-modules $X_{\check G}$, it is easy to show that $\beta _{\check P}^{\rm WD}$ is proper. The following summarizes properties of the spaces just introduced (which are proved using methods similar to those in § 2).

Let $\check P\subset \check G$ be a parabolic subgroup with Levi quotient $\check M$.

1. (i) The space $X_{\check G}^{\rm WD}$ is reduced and a local complete intersection. (This follows from [Reference Dat, Helm, Kurinczuk and MossDHKM20, Theorem 4.1]. See also [Reference ZhuZhu20, Proposition 3.1.6].)

2. (ii) The morphism $\alpha _{\check P}^{\rm WD}:[{\bf X}_{\check P}^{\rm WD}/\check P]\rightarrow [X_{\check M}^{\rm WD}/\check M]$ has finite Tor dimension. (This follows from [Reference ZhuZhu20, Lemma 3.3.1].)

3. (iii) There is a morphism $X_{\check M}^{\rm WD}/\hspace {-.1cm}/\check M\rightarrow X_{\check G}^{\rm WD}/\hspace {-.1cm}/\check G$ making the diagram

   
   commutative. (This is the commutative diagram [Reference ZhuZhu20, (3.10)]. The morphism can easily be constructed using the morphism $X_{\check M}^{\rm WD}\rightarrow \mathbf {X}^{\rm WD}_{\check P}$ induced by the choice of a splitting of $\check P\rightarrow \check M$.)

Remark 3.4 In the case of the space of $(\varphi,N)$-modules $X_{\check G}$ all these properties have been verified in § 2. In relation to (iii) we remark that the morphism

\[ [X_{\check G}/\check G]\longrightarrow X_{\check G}/\hspace{-.1cm}/\check G \]

is just the morphism $\bar \chi$ from (3.1), that is, the GIT quotient $X_{\check G}/\hspace {-.1cm}/\check G$ agrees with the adjoint quotient $\check G/\hspace {-.1cm}/\check G$. This can be seen as follows. The morphism $\varphi \mapsto (\varphi,0)$ defines a closed embedding $\check G\hookrightarrow X_{\check G}$ which is the inclusion of an irreducible component. As $\check G$ is reductive and $C$ has characteristic zero, the category of $\check G$-representations is semi-simple and we obtain a closed embedding

\[ \check G/\hspace{-.1cm}/\check G\longrightarrow X_{\check G}/\hspace{-.1cm}/\check G. \]

As source and target are reduced (as $\check G$ and $X_{\check G}$ are) it is enough to show that the morphism is bijective. This comes down to proving that for $(\varphi,N)\in X_{\check G}(k)$, for an algebraically closed field $k$, there exists $\varphi '\in \check G(k)$ such that

\[ \overline{\check G\cdot (\varphi',0)}\cap \overline{\check G\cdot (\varphi,N)}\neq \emptyset. \]

By (the proof of) Lemma 2.5 we may assume that $\varphi \in \check B$ and $N\in \operatorname {Lie}\check B$ for some Borel $\check B\subset \check G$. Let $\mathbb {G}_m$ act on $X_{\check G}$ by the sum of the positive roots. Then the closure of $\mathbb {G}_m\cdot (\varphi,N)$ contains in addition the point $(\varphi ',0)$ for some $\varphi '\in \check G$ such that $\varphi$ and $\varphi '$ have the same image in the adjoint quotient $\check G/\hspace {-.1cm}/\check G$.

Let us write $\mathfrak {Z}(G)$ for the Bernstein center of the category ${\rm Rep}(G)$. Given a Bernstein component $\Omega$ of ${\rm Rep}(G)$, we denote its center by $\mathfrak {Z}_\Omega (G)$. Moreover, we denote by

\[ \mathcal{Z}(\check G)=\Gamma(X_{\check G}^{\rm WD}/\hspace{-.1cm}/\check G,\mathcal{O}_{X_{\check G}^{\rm WD}/\hspace{-.1cm}/\check G}) \]

the ring of functions on the GIT quotient $X_{\check G}^{\rm WD}/\hspace {-.1cm}/\check G$. If $X\subset X_{\check G}^{\rm WD}$ is a connected component, we write $\mathcal {Z}_X(\check G)$ for the ring of functions on the GIT quotient $X/\hspace {-.1cm}/\check G$.