2.1. Base change to Zastava

In this section, we interpret Whittaker coefficients of Eisenstein series as twisted cohomology of the Zastava space Z.

The fiber product has a stratification indexed by the Weyl group, determined by the generic relative position of two flags. Let $j: Z \hookrightarrow Z'$ be the open inclusion of the locus where the two flags are generically transverse, called the Zastava space.

Consider the compositions

$$\begin{align*}q_{Z'}: Z' \rightarrow {\mathrm{Bun}}_B \rightarrow {\mathrm{Bun}}_T \quad \text{and} \quad \chi_{Z'}: Z' \rightarrow {\mathrm{Bun}}_{N^-}^{\omega^{-\rho}} \rightarrow \mathbf{A}^1\end{align*}$$

and let and be their restrictions to Z.

Proposition 2.1. There is an isomorphism

(2.1) $$ \begin{align} {\mathrm{Hom}}({\mathrm{Whit}}, {\mathrm{Eis}}_! K^{\lambda})^*[-d_G] \simeq {\mathrm{Hom}}(\chi_Z^* D\exp, q_Z^! DK^{\lambda})[d_T + d^0].\end{align} $$

Proof. We cannot directly apply adjunction to calculate Whittaker coefficients of Eisenstein series because $\operatorname {\mathrm {Eis}}_!$ is a left not right adjoint. It is shown in [Reference Færgeman and Raskin10] and [Reference Nadler and Taylor22] that the shifted Whittaker functional $\operatorname {\mathrm {Hom}}(\operatorname {\mathrm {Whit}}, -)[d_B^0]$ on nilpotent sheaves commutes with Verdier duality D. This allows us to exchange for . Then apply adjunction and base change to reduce to a calculation on the fiber product $Z'$ .

$$\begin{align*}{\mathrm{Hom}}({\mathrm{Whit}}, {\mathrm{Eis}}_! K^{\lambda})^*[-2d_B^0] \simeq {\mathrm{Hom}}({\mathrm{Whit}}, {\mathrm{Eis}}_*DK^{\lambda}) \simeq {\mathrm{Hom}}(\chi_{Z'}^* D\exp, q_{Z'}^!DK^{\lambda})\end{align*}$$

Finally, by Equation (3.5) of [Reference Arinkin and Gaitsgory1], restriction to the open generically transverse locus Z does not change the calculation. More precisely, the map

$$\begin{align*}{\mathrm{Hom}}(\chi_{Z'}^* D\exp, q_{Z'}^!DK^{\lambda}) \xrightarrow{\sim} {\mathrm{Hom}}(\chi_Z^* D\exp, q_Z^! DK^{\lambda})\end{align*}$$

is an isomorphism. For the shifts, use (1.4) and $d_G + d_T + d^0 = 2d_B^0$ .

2.2. Pushforward to the configuration space

In this section, we recall how to factor the projection $q_Z: Z^{\lambda } \rightarrow \operatorname {\mathrm {Bun}}_T^{\lambda }$ through the configuration space $X^{\lambda }$ of positive coweight valued divisors of total degree $\lambda $ . Hence, we obtain a description of the $\lambda $ -graded piece of Proposition 2.1 as cohomology of a certain perverse sheaf $\Upsilon ^{\lambda }_{\sigma }$ on $X^{\lambda }$ .

Let $(F, F^-, E) \in Z^{\lambda }$ be a point in the $\lambda $ connected component of Zastava space – that is, a G-bundle E equipped with generically transverse $B, B^-$ -reductions $F, F^-$ , such that F has degree $-\lambda - 2(g-1)\rho $ , and $F^- \times _{B^-} T$ is identified with $\omega ^{-\rho }$ . For each dominant weight $\check {\mu }$ , the Plucker description gives maps

(2.2) $$ \begin{align} F^{\check{\mu}} \rightarrow E^{\check{\mu}} \rightarrow (F^-)^{\check{\mu}} \simeq \omega^{-\langle \check{\mu}, \rho \rangle}.\end{align} $$

Here, is a line bundle and is the vector bundle associated to the simple G-module of highest weight $\check {\mu }$ .

By the generic transversality condition, the composition (2.2) is nonzero map of line bundles, so $\lambda $ is a non-negative coweight. For each point in the Zastava space, there is a unique positive coweight valued divisor $\underline {x} \cdot \underline {\lambda } \in X^{\lambda }$ such that (2.2) factors through an isomorphism $F^{\check {\mu }}(\langle \underline {x} \cdot \underline {\lambda }, \check {\mu } \rangle ) \simeq \omega ^{-\langle \check {\mu }, \rho \rangle }$ . Since G is assumed simply connected, we can write $\lambda = \sum n_i \alpha _i$ as a sum of simple coroots and $X^{\lambda } = \prod X^{(n_i)}$ as a product of symmetric powers of the curve. Therefore, $q_Z$ factors through a map $\pi $ to the configuration space followed by the Abel-Jacobi map,

$$\begin{align*}q_Z : Z^{\lambda} \xrightarrow{\pi} X^{\lambda} \xrightarrow{{\mathrm{AJ}}} {\mathrm{Bun}}_T^{\lambda}, \qquad (E, F, F^-) \mapsto \underline{x} \cdot \underline{\lambda} \mapsto \omega^{-\rho}(-\underline{x} \cdot \underline{\lambda}) \simeq F \times_B T.\end{align*}$$

Let $\lambda $ be a coweight and . Let , an $\check {\mathfrak {n}}$ -local system on X. The Chevalley complex on the coweight graded Ran space gives a $\prod S_{n_i}$ equivariant perverse sheaf $A_{X^n}$ on $\prod X^{n_i}$ . Let $\operatorname {\mathrm {sym}}^{\lambda }: X^n \rightarrow X^{\lambda }$ be the partial symmetrization map. There is a certain canonical summand $\Upsilon _{\sigma }^{\lambda } \subset (\operatorname {\mathrm {sym}}^{\lambda }_* A_{X^n})^{\prod S_{n_i}}$ whose stalk at $\underline {x} \cdot \underline {\lambda } \in X^{\lambda }$ is

(2.3) $$ \begin{align} (\Upsilon^{\lambda}_{\sigma})_{\underline{x} \cdot \underline{\lambda}} \simeq \bigotimes C_{\bullet}(\check{\mathfrak{n}}_{\sigma})^{\lambda_i}_{x_i};\end{align} $$

see Section 3.1 of [Reference Braverman and Gaitsgory6] and Section 4 of [Reference Raskin24]. (The definition of $\Upsilon _{\sigma }^{\lambda }$ involves the Chevalley differential, but the associated graded of $\Upsilon _{\sigma }^{\lambda }$ with respect to the Cousin filtration is easier to describe; see Section 3.3 of [Reference Braverman and Gaitsgory6].)

Proposition 2.3. There is an isomorphism

(2.4) $$ \begin{align}{\mathrm{Hom}}(\chi_Z^* D\exp, q_Z^! DK^{\lambda})[d_T + d^0] \simeq \Gamma(X^{\lambda}, \Upsilon_{\sigma}^{\lambda}).\end{align} $$

Proof. Pushing forward to the configuration space $X^{\lambda }$ , the left of (2.4) becomes

$$\begin{align*}{\mathrm{Hom}}(\pi_! \chi_Z^* D \exp, {\mathrm{AJ}}^!DK^{\lambda})[d_T + d^0] \simeq \Gamma(\Upsilon^{\lambda} \otimes ({\mathrm{AJ}}^* K^{\lambda})^*)[d_T + d^{\lambda}] \simeq \Gamma(X^{\lambda}, \Upsilon^{\lambda}_{\sigma}).\end{align*}$$

We used that the configuration space $X^{\lambda }$ is smooth, so the dualizing sheaf is a rank 1 local system. And we used Theorem 4.6.1 of [Reference Raskin24], which says that

$$\begin{align*}D \pi_! \chi_{Z}^* D \exp \simeq \pi_*\chi_Z^! \exp \simeq \Upsilon^{\lambda}[d^{\lambda} - d^0].\end{align*}$$

Here, $d^{\lambda } - d^0 = \dim Z^{\lambda }$ , and $\Upsilon ^{\lambda }$ has stalks $\Upsilon ^{\lambda }_{\underline {x} \cdot \underline {\lambda }} \simeq \bigotimes C_{\bullet }(\check {\mathfrak {n}})^{\lambda _i}.$

Under class field theory (1.1), the stalks of $\operatorname {\mathrm {AJ}}^*K^{\lambda }$ are

$$\begin{align*}({\mathrm{AJ}}^* K^{\lambda})_{\underline{\lambda} \cdot \underline{x}} \simeq \left(\bigotimes \sigma^{-\lambda_i}_{x_i}\right)[d_T + d^{\lambda}]\end{align*}$$

and its $*$ -pullback to $\prod X^{n_i}$ is the $\prod S_{n_i}$ equivariant rank 1 local system $\boxtimes (\sigma ^{-\alpha _i})^{\boxtimes n_i}$ . By the projection formula, $\Upsilon ^{\lambda } \otimes (\operatorname {\mathrm {AJ}}^* K^{\lambda })^* \simeq \Upsilon _{\sigma }^{\lambda }$ .

Combining Propositions 2.1 and 2.3 shows Whittaker coefficients of Eisenstein series is graded dual to global sections of $\Upsilon _{\sigma }$ on the configuration space.

2.3. The chiral enveloping algebra as a Chevalley complex

The local system $\check {\mathfrak {n}}_{\sigma }$ determines a Lie* algebra on the Ran space. Its Lie algebra homology is a factorization algebra, related to $\Upsilon _{\sigma }$ by partial symmetrization (2.6).

A sheaf on the Ran space of X is a collection of sheaves $A_{X^I}$ on each power of the curve $X^I$ , together with compatibility isomorphisms for $!$ -restrictions along partial diagonal maps; see Section 2.1 of [Reference Francis and Gaitsgory8] for the precise definition. Recall from Section 1.2.1 of [Reference Francis and Gaitsgory8] that the category of sheaves on the Ran space admits two tensor products with a map $\otimes ^* \rightarrow \otimes ^{\operatorname {\mathrm {ch}}}$ between them.

Pushing forward along the main diagonal $\Delta : X \rightarrow \operatorname {\mathrm {Ran}}$ , we can regard $\Delta _* \check {\mathfrak {n}}_{\sigma } \in \operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Ran}})$ as a Lie algebra for the $*$ -tensor product. Restricting to $X^2$ , the Lie* bracket $(\Delta _* \check {\mathfrak {n}}_{\sigma } \otimes ^* \Delta _* \check {\mathfrak {n}}_{\sigma })_{X^2} \simeq \check {\mathfrak {n}}_{\sigma } \boxtimes \check {\mathfrak {n}}_{\sigma } \rightarrow (\Delta _* \check {\mathfrak {n}}_{\sigma })_{X^2} \simeq \Delta _* \check {\mathfrak {n}}_{\sigma }$ comes by adjunction from the Lie bracket.

Let be Lie algebra homology of $\Delta _* \check {\mathfrak {n}}_{\sigma }$ with respect to the $*$ -tensor product, viewed by the forgetful functor as a cocommutative coalgebra with respect to the $\operatorname {\mathrm {ch}}$ -tensor product. Proposition 6.1.2 of [Reference Francis and Gaitsgory8] says that A corresponds to the chiral enveloping algebra of $\Delta _* \check {\mathfrak {n}}_{\sigma }$ under the equivalence between factorization and chiral algebras.

The Chevalley complex $A = \bigoplus A^{\lambda }$ is coweight graded because $\operatorname {\mathrm {Sym}} (\check {\mathfrak {n}}_{\sigma }[1])$ is coweight graded and because the Chevalley differential preserves the grading. Choose a coweight $\lambda $ and let . The sheaf $A^{\lambda }_{X^n}$ on $X^n$ is $S_n$ -equivariant and perverse. Symmetrize it along $\operatorname {\mathrm {sym}}: X^n \rightarrow X^{(n)}$ to get a perverse sheaf $(\operatorname {\mathrm {sym}}_*A^{\lambda }_{X^n})^{S_n}$ on the nth symmetric power. (In other words, we pushed forward $A^{\lambda }_{X^n}$ from the stack quotient $X^n/S_n$ to the coarse quotient $X^{(n)}$ .)

Now we describe a certain canonical summand $A^{\lambda }_{X^{(n)}} \subset (\operatorname {\mathrm {sym}}_*A^{\lambda }_{X^n})^{S_n}$ defined in Section 3 of [Reference Braverman and Gaitsgory6]. Let $X_i^{(n)} \subset X^{(n)}$ be the space of effective degree n divisors supported at exactly i points. The $!$ -restriction of $(\operatorname {\mathrm {sym}}_*A^{\lambda }_{X^n})^{S_n}$ to $X_i^{(n)}$ is a local system whose stalk at a divisor $\underline {n} \cdot \underline {x} \in X_i^{(n)}$ is given by

$$\begin{align*}({\mathrm{sym}}_*A^{\lambda}_{X^n_i})^{S_n}_{\underline{n} \cdot \underline{x}} \simeq \bigoplus_{\lambda = \sum \lambda_j} \bigotimes C_{\bullet}(\mathfrak{n}_{\sigma})^{\lambda_j}_{x_j}.\end{align*}$$

The $!$ -restriction of $A^{\lambda }_{X^{(n)}}$ to $X^{(n)}_i \subset X^{(n)}$ is the summand whose stalks are

(2.5) $$ \begin{align} (A^{\lambda}_{X^{(n)}_i})_{\underline{n} \cdot \underline{x}} \simeq \bigoplus_{\substack{\lambda = \sum \lambda_j, \\ \langle \check{\rho}, \lambda_j \rangle = n_j}} \bigotimes C_{\bullet}(\check{\mathfrak{n}}_{\sigma})_{x_j}^{\lambda_j}.\end{align} $$

By Section 11.6 of [Reference Braverman and Gaitsgory6], the pushforward of $\Upsilon _{\sigma }^{\lambda }$ – see (2.3) – along the partial symmetrization map $^{\lambda }\operatorname {\mathrm {sym}}:X^{\lambda } \rightarrow X^{(n)}$ is

(2.6) $$ \begin{align}^{\lambda}{\mathrm{sym}}_*\Upsilon_{\sigma}^{\lambda} \simeq A^{\lambda}_{X^{(n)}}.\end{align} $$

2.4. Factorization homology

In this section, we review, following [Reference Braverman and Gaitsgory6], how factorization homology of can be computed as cohomology on the symmetric power $X^{(n)}$ , where .

Let $\operatorname {\mathrm {FSet}}$ be the category whose objects are finite nonempty sets and whose morphisms are surjective maps. For each surjection $J \twoheadrightarrow I$ , there is a partial diagonal map $\Delta : X^I \rightarrow X^J$ . A sheaf on the Ran space comes with isomorphisms $A_{X^I} \simeq \Delta ^!A_{X^J}$ , so adjunction gives maps $\Delta _* A_{X^I} \rightarrow A_{X^J}$ . Factorization homology is defined in Section 6.3.3 of [Reference Francis and Gaitsgory8] or Section 4.2.2 of [Reference Beilinson and Drinfeld4] as the colimit over these maps

$$\begin{align*}\Gamma({\mathrm{Ran}}, A) \simeq \underset{{\mathrm{FSet}}^{{\mathrm{op}}}}{{\mathrm{colim}}} \Gamma(A_{X^I}).\end{align*}$$

The following proposition is stated in 11.6 of [Reference Braverman and Gaitsgory6], and below, we fill in the proof using the Cousin filtration and ideas from Section 4.2 of [Reference Beilinson and Drinfeld4].

Proposition 2.4. The cohomology of $\Upsilon _{\sigma }$ – see (2.3) – is the factorization homology of the Chevalley complex,

(2.7) $$ \begin{align} \bigoplus_{\lambda} \Gamma(X^{\lambda}, \Upsilon_{\sigma}^{\lambda}) \simeq \Gamma({\mathrm{Ran}}, A).\end{align} $$

Proof. Equation (2.6) relates $\Upsilon _{\sigma }$ to the symmetrization of A. Thus, it suffices to show that

(2.8) $$ \begin{align}\Gamma(X^{\lambda}, \Upsilon_{\sigma}^{\lambda}) \simeq \Gamma(A^{\lambda}_{X^{(n)}}) \rightarrow \Gamma(A^{\lambda}_{X^n}) \rightarrow \Gamma({\mathrm{Ran}}, A^{\lambda})\end{align} $$

is an isomorphism for . Indeed, we will prove that (2.8) is compatible with the Cousin filtration and that it induces an isomorphism on the associated graded pieces.

Consider the filtration on (2.8) whose $\leq i$ th filtered piece consists of sections supported on the partial diagonals of dimensions $\leq i$ . The ith graded piece is

(2.9) $$ \begin{align} \Gamma(A^{\lambda}_{X^{(n)}_i}) \rightarrow \Gamma(A^{\lambda}_{X^n_i}) \rightarrow \underset{{\mathrm{FSet}}^{{\mathrm{op}}}}{{\mathrm{colim}}} \Gamma(A^{\lambda}_{X^I_i}) \simeq {\mathrm{gr}}_i \Gamma({\mathrm{Ran}}, A^{\lambda}).\end{align} $$

Here, $A^{\lambda }_{X_i^{(n)}}$ is the !-restriction of $A^{\lambda }_{X^{(n)}}$ to the space $X_i^{(n)} \subset X^{(n)}$ of effective degree n divisors supported at exactly i points. Similarly, $A^{\lambda }_{X_i^I}$ is the !-restriction of $A^{\lambda }_{X^I}$ to the space $X_i^I \subset X^I$ of I-tuples supported at exactly i points.

The symmetric group $S_i$ acts freely on the space $X^i_i \subset X^i$ of distinct i-tuples of points. By Section 4.2.3 of [Reference Beilinson and Drinfeld4], the ith graded piece of the factorization homology of $A^{\lambda }$ is $\operatorname {\mathrm {gr}}_i \Gamma (\operatorname {\mathrm {Ran}}, A^{\lambda }) \simeq \Gamma (A^{\lambda }_{X^i_i})_{S_i}$ .

The connected components of $X^{(n)}_i$ are indexed by partitions $\underline {n} = n_1 + \dots n_i$ . Also, the local system $A^{\lambda }_{X^i_i}$ splits as a direct sum indexed by such partitions; see 6.4.9 of [Reference Francis and Gaitsgory8]. Restricting (2.9) to the connected component $X^{(\underline {n})}_i \subset X^{(n)}_i$ indexed by a certain partition,

$$\begin{align*}\Gamma(A^{\lambda}_{X^{(\underline{n})}_i}) \rightarrow {\mathrm{gr}}_i \Gamma({\mathrm{Ran}}, A^{\lambda}) \simeq \Gamma(A^{\lambda}_{X^i_i})_{S_i}\end{align*}$$

is an isomorphism onto the corresponding summand of $\Gamma (A^{\lambda }_{X^i_i})_{S_i}$ by (2.5). Summing over partitions shows that the ith graded piece of (2.8) is an isomorphism.

Since the factorization algebra corresponds to the chiral enveloping algebra $U(\check {\mathfrak {n}}_{\sigma })$ , Beilinson and Drinfeld’s formula for chiral homology of an enveloping algebra – see Theorem 4.8.1.1 of [Reference Beilinson and Drinfeld4] or 6.4.4 of [Reference Francis and Gaitsgory8] – says

(2.10) $$ \begin{align} \Gamma({\mathrm{Ran}}, A) \simeq C_{\bullet}(\Gamma(X, \check{\mathfrak{n}}_{\sigma})).\end{align} $$

2.5. Deformation theory

In this section, we show that

(2.11) $$ \begin{align} C_{\bullet}(\Gamma(X, \check{\mathfrak{n}}_{\sigma})) \simeq \mathcal{O}({\mathrm{Loc}}_{\check{N}}^{\sigma})^*,\end{align} $$

Lie algebra homology of the shifted tangent complex equals the graded dual ring of functions on $\operatorname {\mathrm {Loc}}_{\check {N}}^{\sigma }$ . Deformation theory says that $C_{\bullet }(\Gamma (X, \check {\mathfrak {n}}_{\sigma })) \simeq \Gamma ^{\operatorname {\mathrm {IndCoh}}}(\omega _{(\operatorname {\mathrm {Loc}}_{\check {N}}^{\sigma })^{\wedge }})$ is global sections of the dualizing sheaf on the formal completion at $\sigma $ . Using the structure of $\operatorname {\mathrm {Loc}}_{\check {N}}^{\sigma }$ described below, we recover the graded dual ring of functions on $\operatorname {\mathrm {Loc}}_{\check {N}}^{\sigma }$ , not just its completion, from $\Gamma ^{\operatorname {\mathrm {IndCoh}}}(\omega _{(\operatorname {\mathrm {Loc}}_{\check {N}}^{\sigma })^{\wedge }})$ .

First, we show that $\operatorname {\mathrm {Loc}}_{\check {N}}^{\sigma } \simeq \operatorname {\mathrm {Loc}}_{\check {N}}^{\sigma , x}/\check {N}$ is the quotient by a unipotent group of an affine derived scheme with a contracting $\mathbf {G}_m$ -action. Let (respectively, ) be the Betti moduli of $\check {B}$ (respectively, $\check {T}$ ) local systems trivialized at a point x. Let be the moduli of $\check {B}$ -local systems with underlying $\check {T}$ -local system identified with $\sigma $ , plus a $\check {T}$ -reduction at x.

Since $\check {T}$ is abelian, it acts by automorphisms on $\sigma \in \operatorname {\mathrm {Loc}}_{\check {T}}$ so there is a canonical lift $\sigma \in \operatorname {\mathrm {Loc}}_{\check {T}}^x$ . We also sometimes regard $\sigma $ as a point in $\operatorname {\mathrm {Loc}}_{\check {N}}^{\sigma , x}$ via the inclusion $\check {T} \subset \check {B}$ .

Let $\check {B}$ act on $\operatorname {\mathrm {Loc}}_{\check {B}}^x$ by changing the trivialization at x, equivalently by the adjoint action on $\check {B}^{2g} \times _{\check {B}} 1$ . Restricting the adjoint action along $\check {\rho }$ gives a $\mathbf {G}_m$ -action that contracts $\check {B}$ to $\check {T}$ . Thus, we expect a $\mathbf {G}_m$ -action that contracts $\operatorname {\mathrm {Loc}}_{\check {B}}^x$ to $\operatorname {\mathrm {Loc}}_{\check {T}}^x$ , as is made precise below.

Proof. We argue in the Betti setting, but the restricted and de Rham versions also follow by (1.3). First, rewrite

(2.12) $$ \begin{align} {\mathrm{Loc}}_{\check{N}}^{\sigma, x} \simeq {\mathrm{Loc}}_{\check{B}}^x \times_{{\mathrm{Loc}}_{\check{T}}^x} \sigma \simeq \check{B}^{2g} \times_{\check{T}^{2g} \times_{\check{T}} \check{B}} \sigma \simeq (\check{B}^{2g} \times_{\check{T}^{2g}} \sigma) \times_{\check{B} \times_{\check{T}} 1} 1 \simeq \operatorname{\mathrm{Spec}} (R' \otimes_S k).\end{align} $$

The contracting $\check {\rho }$ -action induces non-negative gradings on the classical rings and . Since $\check {N}$ is smooth, the augmentation module $k \simeq S/S_{> 0}$ admits a finite graded resolution by free S-modules, with all but one term shifted into strictly positive $\check {\rho }$ -gradings. Therefore, $R \simeq R' \otimes _S k$ is a finite type non-negatively graded ring and $\sigma \simeq \operatorname {\mathrm {Spec}} R/R_{>0}$ .

Now we review some derived deformation theory. Let $Y^{\wedge }$ be the formal completion of a derived stack Y at a point $\sigma $ . The shifted tangent bundle $T_{\sigma }Y[-1]$ is a DG Lie algebra whose enveloping algebra is endomorphisms of the skyscraper at $\sigma $ . By Chapter 7 of [Reference Gaitsgory and Rozenblyum17] or Remark 2.4.2 of [Reference Lurie21], there is an equivalence

$$\begin{align*}{\mathrm{Mod}}(T_{\sigma}Y[-1]) \simeq {\mathrm{IndCoh}}(Y^{\wedge})\end{align*}$$

between Lie algebra modules for the shifted tangent complex and indcoherent sheaves on the formal completion. Let $p: Y^{\wedge } \rightarrow \operatorname {\mathrm {pt}}$ be the map to a point. By Chapter 7, Section 5.2 of [Reference Gaitsgory and Rozenblyum17], the trivial $T_{\sigma }Y[-1]$ -module corresponds to the dualizing sheaf $\omega _{Y^{\wedge }} \simeq p^! k \in \operatorname {\mathrm {IndCoh}}(Y^{\wedge })$ . Moreover, Lie algebra homology corresponds to global sections

(2.13) $$ \begin{align} C_{\bullet}(T_{\sigma}Y[-1]) \simeq \Gamma^{{\mathrm{IndCoh}}}(\omega_{Y^{\wedge}}).\end{align} $$

Suppose $Y^{\wedge } \simeq \operatorname {\mathrm {Spec}} R$ is the spectrum of an Artinian local ring R. By properness, $p^!$ is right adjoint to $p_*^{\operatorname {\mathrm {IndCoh}}}$ . Therefore, the dualizing complex $\omega _{Y^{\wedge }} \simeq R^*$ is the linear dual of R viewed as an R-module.

Suppose $Y^{\wedge } \simeq \operatorname {\mathrm {Spf}} R^{\wedge } \simeq \operatorname {\mathrm {colim}} Y_n$ where $Y_n \simeq \operatorname {\mathrm {Spec}} R/\mathfrak {m}^n$ and let $i_n: Y_n \rightarrow Y^{\wedge }$ . Since $Y_n \rightarrow Y_{n+1}$ is proper, $\operatorname {\mathrm {IndCoh}}(Y^{\wedge })$ is the colimit under $*$ -pushforward of $\operatorname {\mathrm {IndCoh}}(Y_n)$ ; see Chapter 1, Proposition 2.5.7 of [Reference Gaitsgory and Rozenblyum16]. The dualizing sheaf can be written as a colimit, $\omega _{Y^{\wedge }} \simeq \operatorname {\mathrm {colim}} i_{n*}^{\operatorname {\mathrm {IndCoh}}} \omega _{Y_n}$ ; see Chapter 7, Corollary 5.3.3 of [Reference Gaitsgory and Rozenblyum17]. Since $\Gamma ^{\operatorname {\mathrm {IndCoh}}}(Y^{\wedge }, -)$ is continuous, it follows that

(2.14) $$ \begin{align} \Gamma^{{\mathrm{IndCoh}}}(\omega_{Y^{\wedge}}) \simeq {\mathrm{colim}} ((R/\mathfrak{m}^n)^*) \simeq (R^{\wedge})^*\end{align} $$

is the topological dual of the completed local ring $R^{\wedge }$ . In this case, Equation (2.13) is Corollary 5.2 of [Reference Hinich18].

Proof. First, suppose R is classical and choose homogeneous generators $f_1, \dots f_r \in R$ . Let d be the maximum of their degrees, so $R_{\geq dn} \subset (f_1, \dots f_r)^n \subset R_{\geq n}$ . Therefore, the graded dual $R^*$ (linear functionals that vanish on some $R_{\geq n}$ ) equals the topologogical dual $(R^{\wedge })^*$ (linear functionals that vanish on some $(f_1, \dots f_r)^n$ ).

Now suppose that R is derived. The finite type assumption means that after taking cohomology, $H^{\bullet }(R)$ is a finitely generated module over $H^0(R)$ , a finitely generated graded classical ring. Choose a finite collection of homogeneous elements $f_1, \dots f_r \in R$ whose images generate $H^0(R)$ .

The formal completion is the topological ring

$$\begin{align*}R^{\wedge} \simeq R \otimes_{k[f_1, \dots f_r]} k[[f_1, \dots f_r]] \simeq \lim_n R \otimes_{k[f_1, \dots f_r]} (k[f_1, \dots f_r]/k[f_1, \dots f_r]_{> n}).\end{align*}$$

For the first equality, see Section 6.7 of [Reference Gaitsgory and Rozenblyum15]. The second equality uses that fiber products commute with filtered colimits and that $k[[f_1, \dots f_r]] \simeq \lim (k[f_1, \dots f_r]/k[f_1, \dots f_r]_{> n})$ . (The formal completion of a classical positively graded polynomial algebra can be computed using the grading filtration.)

Since $k[f_1, \dots f_r]$ is smooth, $R \otimes _{k[f_1, \dots f_r]}k[f_1, \dots f_r]/k[f_1, \dots f_r]_{>n}$ has finite dimensional cohomology and therefore is concentrated in bounded degrees. Hence, for m sufficiently large, the quotient map factors through

$$\begin{align*}R \rightarrow R/R_{> m} \rightarrow R \otimes_{k[f_1, \dots f_r]}k[f_1, \dots f_r]/k[f_1, \dots f_r]_{>n} \rightarrow R/R_{>n}.\end{align*}$$

Therefore, the formal completion of R can be computed using the grading filtration

$$\begin{align*}R^{\wedge} \simeq \lim_n R \otimes_{k[f_1, \dots f_r]} (k[f_1, \dots f_r]/k[f_1, \dots f_r]_{> n}) \simeq \lim_n R/R_{> n}.\end{align*}$$

Taking the topological dual proves $(R^{\wedge })^* \simeq \operatorname {\mathrm {colim}} ((R/R_{> n})^*) \simeq \bigoplus R_n^* \simeq R^*$ .

The following proposition shows (2.11), completing the final step of (1.5) and the proof of the main theorem.

Proposition 2.7. Lie algebra homology of the shifted tangent complex of equals the graded dual of the ring of functions,

$$\begin{align*}C_{\bullet}(T_{\sigma}Y[-1]) \simeq \mathcal{O}(Y)^*.\end{align*}$$

Proof. Write $\operatorname {\mathrm {Loc}}_{\check {N}}^{\sigma , x} \simeq \operatorname {\mathrm {Spec}} R$ as in Proposition 2.5. Let $\check {N}$ act by changing the $\check {T}$ -reduction at x. Since $\check {T}$ normalizes $\check {N}$ , the quotient $Y \simeq (\operatorname {\mathrm {Spec}} R)/\check {N}$ retains the $\check {\rho }$ -action. The formal completion of Y at $\sigma $ is the inf-scheme $Y^{\wedge } \simeq \operatorname {\mathrm {Spf}}(R^{\wedge })/\exp (\check {\mathfrak {n}})$ , the quotient by the formal group $\exp (\check {\mathfrak {n}})$ .

Deformation theory says

$$\begin{align*}C_{\bullet}(T_{\sigma} Y[-1]) \simeq \Gamma^{{\mathrm{IndCoh}}}(\omega_{Y^{\wedge}}) \simeq ((R^{\wedge})^*)_{\check{\mathfrak{n}}}.\end{align*}$$

The first equality is equation (2.13). For the second equality, we pushed forward the dualizing sheaf $\omega _{Y^{\wedge }}$ in two steps,

$$\begin{align*}Y^{\wedge} \rightarrow {\mathrm{pt}}/\exp(\check{\mathfrak{n}}) \rightarrow {\mathrm{pt}}.\end{align*}$$

The pushforward of $\omega _{Y^{\wedge }}$ to $\operatorname {\mathrm {pt}}/\exp (\check {\mathfrak {n}})$ is an $\check {\mathfrak {n}}$ -module. By proper base change and (2.14), the underlying vector space is $\Gamma ^{\operatorname {\mathrm {IndCoh}}}(\omega _{\operatorname {\mathrm {Spf}} R^{\wedge }}) \simeq (R^{\wedge })^*$ and the $\check {\mathfrak {n}}$ -module structure comes from the $\check {N}$ -action. Further pushing forward along $\operatorname {\mathrm {pt}}/\exp (\check {\mathfrak {n}}) \rightarrow \operatorname {\mathrm {pt}}$ corresponds to taking $\check {\mathfrak {n}}$ -coinvariants, so $\Gamma ^{\operatorname {\mathrm {IndCoh}}}(\omega _{Y^{\wedge }}) \simeq ((R^{\wedge })^*)_{\check {\mathfrak {n}}}$ .

Now we show that $\check {\mathfrak {n}}$ -coinvariants of the topological dual of $R^{\wedge }$ equals the graded dual ring of functions on Y,

$$\begin{align*}((R^{\wedge})^*)_{\check{\mathfrak{n}}} \simeq {\mathrm{colim}} (((R/R_{> n})^*)_{\check{\mathfrak{n}}}) \simeq {\mathrm{colim}} (((R/R_{> n})^{\check{\mathfrak{n}}})^*) \simeq {\mathrm{colim}} (((R^{\check{\mathfrak{n}}} / (R^{\check{\mathfrak{n}}})_{> n})^*) \simeq (R^{\check{N}})^*.\end{align*}$$

The ideal $R_{> n}$ is an $\check {\mathfrak {n}}$ -module because the $\check {\mathfrak {n}}$ -action increases $\check {\rho }$ -weights. For the first equality, Proposition 2.6 says that $(R^{\wedge })^* \simeq \operatorname {\mathrm {colim}} ((R/R_{> n})^*)$ , and coinvariants commutes with colimits. For the second equality, $((R/R_{> n})^*)_{\check {\mathfrak {n}}} \simeq ((R/R_{> n})^{\check {\mathfrak {n}}})^*$ because $R/R_{> n}$ has finite dimensional cohomology. For the third equality, the image of $(R_{> n})^{\check {\mathfrak {n}}} \rightarrow R^{\check {\mathfrak {n}}}$ is concentrated in degrees $> n$ so we get a map $(R/R_{> n})^{\check {\mathfrak {n}}} \rightarrow R^{\check {\mathfrak {n}}}/(R^{\check {\mathfrak {n}}})_{>n}$ . Moreover, since $(R/R_{> n})^{\check {\mathfrak {n}}}$ is concentrated in bounded degrees, for m sufficiently large, the quotient map factors through

$$\begin{align*}R^{\check{\mathfrak{n}}} / (R^{\check{\mathfrak{n}}})_{> m} \rightarrow (R/R_{> n})^{\check{\mathfrak{n}}} \rightarrow R^{\check{\mathfrak{n}}}/(R^{\check{\mathfrak{n}}})_{> n}.\end{align*}$$

For the fourth equality, we used the van Est isomorphism; see Theorem 5.1 of [Reference Hochschild19]. Since $\check {N}$ is unipotent, Lie algebra cohomology $R^{\check {\mathfrak {n}}}$ coincides with group cohomology $R^{\check {N}}$ .

Example 2.8. Let $G = \operatorname {\mathrm {SL}}(2)$ and let $\sigma $ be a $\check {T}$ -local system, viewed as a rank 1 local system using the positive coroot. Then $\sigma $ is regular if and only if it is nontrivial.

If $\sigma $ is regular, then $\operatorname {\mathrm {Loc}}_{\check {N}}^{\sigma } \simeq H^1(X, \sigma )$ is a classical affine scheme because the other cohomologies vanish. The shifted tangent complex $T_{\sigma }\operatorname {\mathrm {Loc}}_{\check {N}}^{\sigma }[-1] \simeq H^1(X, \sigma )[-1]$ is an abelian Lie algebra with enveloping algebra . Lie algebra homology of the shifted tangent complex is

$$\begin{align*}k \otimes_U k \simeq {\mathrm{Sym}} H^1(X, \sigma) \simeq \mathcal{O}({\mathrm{Loc}}_{\check{N}}^{\sigma})^*.\end{align*}$$

If $\sigma $ is trivial, then $C_{\bullet }(T_{\sigma } \operatorname {\mathrm {Loc}}_{\check {N}}[-1]) \simeq \operatorname {\mathrm {Sym}}(H^2(X)[-1] \oplus H^1(X) \oplus H^0(X)[1])$ is the graded dual ring of functions on $\operatorname {\mathrm {Loc}}_{\check {N}} \simeq H^2(X)[-1] \times H^1(X) \times \operatorname {\mathrm {pt}}/H^0(X)$ .