1 Introduction
Let $(V,Q)$ be a quadratic space over $\mathbb {Q}$ of signature $(p,q)$ , and let G be its orthogonal group. Let $\mathbb {D}$ be the space of oriented negative q-planes in $V(\mathbb {R})$ and $\mathbb {D}^+$ one of its connected components. It is a Riemannian manifold of dimension $pq$ and an open subset of the Grassmannian. The Lie group $G(\mathbb {R})^+$ is the connected component of the identity and acts transitively on $\mathbb {D}^+$ . Hence, we can identify $\mathbb {D}^+$ with $G(\mathbb {R})^+ /K $ , where K is a compact subgroup of $G(\mathbb {R})^+$ and is isomorphic to $\operatorname {SO}(p)\times \operatorname {SO}(q)$ . Moreover, let L be a lattice in $ V(\mathbb {Q}),$ and let $\Gamma $ be a torsion-free subgroup of $G(\mathbb {R})^+$ preserving L.
For every vector v in $V(\mathbb {R})$ such that $Q(v,v)>0$ , there is a totally geodesic submanifold $\mathbb {D}^+_v$ of codimension q consisting of all the negative q-planes that are orthogonal to v. Let $\Gamma _v$ denote the stabilizer of v in $\Gamma $ . We can view $ \Gamma _v \backslash \mathbb {D}^+$ as a rank q vector bundle over $\Gamma _v \backslash \mathbb {D}_v^+$ , so that the natural embedding $\Gamma _v \backslash \mathbb {D}_v^+ $ in $ \Gamma _v \backslash \mathbb {D}^+$ is the zero section. In [Reference Kudla and Millson6], Kudla and Millson constructed a closed $G(\mathbb {R})^+$ -invariant differential form
where $G(\mathbb {R})^+$ acts on the Schwartz space $\mathscr {S}(V(\mathbb {R}))$ from the left by and on $\Omega ^q(\mathbb {D}^+) \otimes \mathscr {S}(V(\mathbb {R}))$ from the right by . In particular, $\varphi _{KM}(v)$ is a $\Gamma _v$ -invariant form on $\mathbb {D}^+$ . The main property of the Kudla–Millson form is its Thom form property: if $\omega $ in $\Omega _c^{pq-q}(\Gamma _v \backslash \mathbb {D}^+)$ is a compactly supported form, then
Another way to state it is to say that in cohomology, we have
where $\operatorname {PD}(\Gamma _v \backslash \mathbb {D}^+_v)$ denotes the Poincaré dual class to $\Gamma _v \backslash \mathbb {D}^+_v$ .
1.1 Kudla–Millson theta lift
In order to motivate the interest in the Kudla–Millson form, let us briefly recall how it is used to construct a theta correspondence between certain cohomology classes and modular forms. For simplicity,Footnote 1 assume that $p+q$ is even, and let $\omega $ be the Weil representation of the dual pair $\operatorname {SL}_2(\mathbb {R}) \times G(\mathbb {R})$ in $\mathscr {S}(V(\mathbb {R}))$ . We extend it to a representation in $\Omega ^q(\mathbb {D}^+) \otimes \mathscr {S}(V(\mathbb {R}))$ by acting in the second factor of the tensor product. Building on the work of [Reference Weil11], Kudla and Millson [Reference Kudla and Millson7, Reference Kudla and Millson9] used their differential form to construct the theta series
where $\tau =x+iy$ is in $\mathbb {H}$ and $g_\tau $ is the matrix $\begin {pmatrix} \sqrt {y} & x\sqrt {y}^{-1} \\ 0 & \sqrt {y}^{-1} \end {pmatrix}$ in $\operatorname {SL}_2(\mathbb {R})$ that sends i to $\tau $ by Möbius transformation. This form is $\Gamma $ -invariant, closed and holomorphic in cohomology in the sense that $\frac {\partial }{\partial \overline {\tau }}\Theta _{KM}(\tau )$ is an exact form. Kudla and Millson showed that if we integrate this closed form on a compact q-cycle C in $\mathcal {Z}_q(\Gamma \backslash \mathbb {D}^+)$ , then
is a modular form of weight $\frac {p+q}{2}$ , where
and the special cycles $C_v$ are the images of the composition
Thus, the Kudla–Millson theta series realizes a lift between the (co)-homology of $\Gamma \backslash \mathbb {D}^+$ and the space of weight $\frac {p+q}{2}$ modular forms.
1.2 The result
Let E be a $G(\mathbb {R})^+$ -equivariant vector bundle of rank q over $\mathbb {D}^+$ , and let $E_0$ be the image of the zero section. By the equivariance, we also have a vector bundle $\Gamma _v \backslash E$ over $\Gamma _v \backslash \mathbb {D}^+$ . The Thom class of the vector bundle is a characteristic class $\operatorname {Th}(\Gamma _v \backslash E)$ in $ H^{q}(\Gamma _v \backslash E,\Gamma _v \backslash (E-E_0))$ defined by the Thom isomorphism (see Section 3.6). A Thom form is a form representing the Thom class. It can be shown that the Thom class is also the Poincaré dual class to $\Gamma _v \backslash E_0$ . Let $s_v \colon \Gamma _v \backslash \mathbb {D}^+ \longrightarrow \Gamma _v \backslash E$ be a section whose zero locus is $\Gamma _v \backslash \mathbb {D}_v^+$ , then
Viewing it as a class in $H^q(\Gamma _v \backslash \mathbb {D}^+)$ it is the Poincaré dual class of $\Gamma _v \backslash \mathbb {D}_v^+$ . Since the Poincaré dual class is unique, property (1.3) implies that
on the level of cohomology.
For arbitrary real oriented metric vector bundles, Mathai and Quillen used the Chern–Weil theory to construct in [Reference Mathai and Quillen10] a canonical Thom form on E. We denote by $U_{MQ}$ the canonical Thom form in $\Omega ^{q}(E)$ of Mathai and Quillen. Since $U_{MQ}$ is $\Gamma $ -invariant, it is also a Thom form for the bundle $\Gamma _v \backslash E$ for every vector v. The main result is the following.
Theorem (Theorem 4.5)
For a natural choice of a bundle E and of a section $s_v$ , we have $\varphi _{KM}(v)= 2^{-\frac {q}{2}}e^{-\pi Q(v,v)} s_v^\ast U_{MQ}$ in $\Omega ^q(\Gamma _v \backslash \mathbb {D}^+).$
The bundle E is the tautological bundle of the Grassmannian $\mathbb {D}^+$ (see Section 3.6), and the section $s_v$ is defined in Section 4.1.
For signature $(2,q)$ , the spaces are Hermitian and the result was obtained by a similar method in [Reference Garcia3] using the work of Bismut–Gillet–Soulé.
1.3 Generalizations
More generally, for a positive nondegenerate r-subspace $U \subset V$ spanned by vectors $v_1, \dots , v_r$ , Kudla and Millson also construct an $rq$ form $\varphi _{KM}(v_1,\dots ,v_r)$ . This form can also be recovered by the Mathai–Quillen formalism (see (3) of Section 5). Furthermore, in [Reference Kudla and Millson7, Reference Kudla and Millson9], they not only construct forms for the symmetric space associated with $\operatorname {SO}(p,q)$ , but also for the Hermitian space associated with $U(p,q)$ . In this case, one should be able to recover their forms using the formalism of superconnections as in [Reference Mathai and Quillen10, Theorem 8.5]. We expect the computations to be closer to the computations done in [Reference Garcia3].
2 The Kudla–Millson form
2.1 The symmetric space $\mathbb {D}$
Let $(V,Q)$ be a rational quadratic space, and let $(p,q)$ be the signature of $V(\mathbb {R})$ . Let $e_1, \dots , e_{p+q}$ be an orthogonal basis of $V(\mathbb {R})$ such that
Note that we will always use letters $\alpha $ and $\beta $ for indices between $1$ and p, and letters $\mu $ and $\nu $ for indices between $p+1$ and $p+q$ . A plane z in $V(\mathbb {R})$ is a negative plane if $Q\big|{}_{z} $ is negative definite. Let
be the set of negative-oriented q-planes in $V(\mathbb {R})$ . For each negative plane, there are two possible orientations, yielding two connected components $\mathbb {D}^+ $ and $ \mathbb {D}^-$ of $\mathbb {D}$ . Let $z_0$ in $\mathbb {D}^+$ be the negative plane spanned by the vectors $e_{p+1}, \dots , e_{p+q}$ together with a fixed orientation. The group $G(\mathbb {R})^+$ acts transitively on $\mathbb {D}^+$ by sending $z_0$ to $gz_0$ . Let K be the stabilizer of $z_0$ , which is isomorphic to $\operatorname {SO}(p)\times \operatorname {SO}(q)$ . Thus, we have an identification
For z in $\mathbb {D}^+$ , we denote by $g_z$ any element of $G(\mathbb {R})^+$ sending $z_0$ to z.
For a positive vector v in $V(\mathbb {R}),$ we define
It is a totally geodesic submanifold of $\mathbb {D}$ of codimension q. Let $\mathbb {D}_v^+$ be the intersection of $\mathbb {D}_v$ with $\mathbb {D}^+$ .
Let z in $\mathbb {D}^+$ be a negative plane. With respect to the orthogonal splitting of $V(\mathbb {R})$ as $z^\perp \oplus z$ , the quadratic form splits as
We define the Siegel majorant at z to be the positive-definite quadratic form
2.2 The Lie algebras $\mathfrak {g}$ and $\mathfrak {k}$
Let
be the Lie algebras of $G(\mathbb {R})^+$ and $K,$ where $\mathfrak {so}(z_0)$ is equal to $\mathfrak {so}(q)$ . The latter is the space of skew-symmetric q by q matrices. Similarly, we have $\mathfrak {so}(z_0^\perp )$ equals $\mathfrak {so}(p)$ . Hence, we have a decomposition of $\mathfrak {k}$ as $\mathfrak {so}(z_0^\perp ) \oplus \mathfrak {so}(z_0)$ that is orthogonal with respect to the Killing form. Let $\epsilon $ be the Lie algebra involution of $\mathfrak {g}$ mapping X to $-X$ . The $+1$ -eigenspace of $\epsilon $ is $\mathfrak {k}$ and the $-1$ -eigenspace is
We have a decomposition of $\mathfrak {g} $ as $\mathfrak {k} \oplus \mathfrak {p}$ and it is orthogonal with respect to the Killing form. We can identify $\mathfrak {p}$ with $\mathfrak {g}/\mathfrak {k}$ . Since $\epsilon $ is a Lie algebra automorphism, we have that
We identify the tangent space of $\mathbb {D}^+$ at $eK$ with $\mathfrak {p}$ and the tangent bundle $T\mathbb {D}^+$ with $G(\mathbb {R})^+ \times _K \mathfrak {p}$ , where K acts on $\mathfrak {p}$ by the $\operatorname {Ad}$ -representation. We have an isomorphism
A basis of $\mathfrak {g}$ is given by the set of matrices
and we denote by $\omega _{ij}$ , its dual basis in the dual space $\mathfrak {g}^\ast $ . Let $E_{ij}$ be the elementary matrix sending $e_i$ to $e_j$ and the other $e_k$ ’s to $0$ . Then $\mathfrak {p}$ is spanned by the matrices
and $\mathfrak {k}$ is spanned by the matrices
2.3 Poincaré duals
Let M be an arbitrary m-dimensional real orientable manifold without boundary. The integration map yields a nondegenerate pairing [Reference Bott and Tu2, Theorem 5.11]
where $H_c(M)$ denotes the cohomology of compactly supported forms on M. This yields an isomorphism between $H^{q}(M)$ and the dual $H_c^{m-q}(M)^\ast =\operatorname {Hom}(H_c^{m-q}(M),\mathbb {R})$ . If C is an immersed submanifold of codimension q in M, then C defines a linear functional on $H_c^{m-q}(M)$ by
Since we have an isomorphism between $H_c^{m-q}(M)^\ast $ and $H^{q}(M)$ , there is a unique cohomology class $\operatorname {PD}(C)$ in $H^q(M)$ representing this functional, i.e.,
for every class $[\omega ]$ in $H_c^{m-q}(M)$ . We call $\operatorname {PD}(C)$ the Poincaré dual class to C, and any differential form representing the cohomology class $\operatorname {PD}(C)$ a Poincaré dual form to C.
2.4 The Kudla–Millson form
The tangent plane at the identity $T_{eK} \mathbb {D}^+ $ can be identified with $\mathfrak {p}$ and the cotangent bundle $(T\mathbb {D}^+)^\ast $ with $G(\mathbb {R})^+ \times _K \mathfrak {p}^\ast $ , where K acts on $\mathfrak {p}^\ast $ by the dual of the $\operatorname {Ad}$ -representation. The basis $e_1, \dots , e_{p+q}$ identifies $V(\mathbb {R})$ with $\mathbb {R}^{p+q}$ . With respect to this basis, the Siegel majorant at $z_0$ is given by
Recall that $G(\mathbb {R})^+$ acts on $\mathscr {S}(\mathbb {R}^{p+q})$ from the left by $(g \cdot f)(v)= f(g^{-1}v)$ and on $\Omega ^q(\mathbb {D}^+) \otimes \mathscr {S}(\mathbb {R}^{p+q})$ from the right by . We have an isomorphism
by evaluating $\varphi $ at the basepoint $eK$ in $G(\mathbb {R})^+/K$ , corresponding to the point $z_0$ in $\mathbb {D}^+$ . We define the Howe operator
by
where $A_{\alpha \mu }$ denotes left multiplication by $\omega _{\alpha \mu }$ . The Kudla–Millson form is defined by applying D to the Gaussian:
Kudla and Millson showed that this form is K-invariant. Hence, by the isomorphism (2.19), we get a form
In particular, since $g^\ast \varphi _{KM}(v)=\varphi _{KM}(g^{-1}v)$ for any $g \in G(\mathbb {R})^+$ , the form is $\Gamma _v$ -invariant and defines a form on $\Gamma _v \backslash \mathbb {D}^+$ . It is also closed and Kudla–Millson prove in [Reference Kudla and Millson8, Proposition 5.2] that it satisfies the Thom form property: for every compactly supported form $\omega $ in $\Omega ^{pq-q}_c(\Gamma _v \backslash \mathbb {D}^+)$ , we have
3 The Mathai–Quillen formalism
We begin by recalling a few facts about principal bundles, connections, and associated vector bundles. For more details, we refer to [Reference Berline, Getzler and Vergne1, Reference Kobayashi and Nomizu5]. The Mathai–Quillen form is defined in Section 3.7 following [Reference Berline, Getzler and Vergne1] (see also [Reference Getzler, Cruzeiro and Zambrini4]).
3.1 K-principal bundles and principal connections
Let K be $\operatorname {SO}(p)\times \operatorname {SO}(q)$ as before, and let P be a smooth principal K-bundle. Let
be the smooth right action of K on P and
the projection map. For a fixed $p $ in P, consider the map
Let $V_pP$ be the image of the derivative at the identity
which is injective. It coincides with the kernel of the differential $d_p\pi $ . A vector in $V_pP$ is called a vertical vector. Using this map, we can view a vector X in $\mathfrak {k}$ as a vertical vector field on P. The space P can a priori be arbitrary, but in our case, we will consider either:
-
(1) P is $G(\mathbb {R})^+$ and $R_k$ the natural right action sending g to $gk$ . Then $P/K$ can be identified with $\mathbb {D}^+$ .
-
(2) P is $G(\mathbb {R})^+ \times z_0$ and the action $R_k$ maps $(g,w)$ to $(gk,k^{-1}w)$ . In this case, $P/K$ can be identified with $G(\mathbb {R})^+ \times _K z_0$ . It is the vector bundle associated with the principal bundle $G(\mathbb {R})^+$ as defined below.
A principal K-connection on P is a $1$ -form $\theta _P $ in $\Omega ^1(P, \mathfrak {k})$ such that:
-
• $\iota _X \theta _P = X$ for any $X $ in $\mathfrak {k}$ ,
-
• $R_k^\ast \theta _P=Ad(k^{-1}) \theta _P \quad $ for any k in K,
where $\iota _X$ is the interior product
and we view X as a vector field on P. Geometrically, these conditions imply that the kernel of $\theta _P$ defines a horizontal subspace of $TP$ that we denote by $HP$ . It is a complement to the vertical subspace, i.e., we get a splitting of $T_pP$ as $V_pP \oplus H_pP$ .
Let $\mathfrak {g}$ be the Lie algebra of $G(\mathbb {R})^+$ , and let $\mathcal {P}$ be the orthogonal projection from $\mathfrak {g}$ on $\mathfrak {k}$ . After identifying $\mathfrak {g}^\ast $ with the space $\Omega ^1(G(\mathbb {R})^+)^{G(\mathbb {R})^+}$ of $G(\mathbb {R})^+$ -invariant forms, we define a natural $1$ -form
called the Maurer–Cartan form, where $X_{ij}$ is the basis of $\mathfrak {g}$ defined earlier and $\omega _{ij}$ its dual in $\mathfrak {g}^\ast $ . After projection onto $\mathfrak {k}$ , we get a form
where we identify $\Omega ^1(G(\mathbb {R})^+, \mathfrak {k})$ with $\Omega ^1(G(\mathbb {R})^+) \otimes \mathfrak {k}$ . A direct computation shows that it is a principal K-connection on P, when P is $G(\mathbb {R})^+$ .
If P is $G(\mathbb {R})^+ \times z_0$ , then the projection
induces a pullback map
The form
is a principal connection on $G(\mathbb {R})^+ \times z_0$ .
3.2 The associated vector bundles
Since $z_0$ is preserved by K, we have an orthogonal K-representation
where we will usually simply write $kw$ instead of ${k\big|{}_{z_0} }w$ . We can consider the associated vector bundle $P \times _K z_0$ which is the quotient of $P \times z_0$ by K, where K acts by sending $(p,w)$ to $ (R_k(p), \rho (k)^{-1}w)$ . Hence, an element $[p,w]$ of $P \times _K z_0$ is an equivalence class where the equivalence relation identifies $(p,w)$ with $(R_k(p), \rho (k)^{-1}w)$ . This is a vector bundle over $P/K$ with projection map sending $[p,w]$ to $\pi (p)$ . Let $\Omega ^i(P/K,P \times _Kz_0)$ be the space of i-forms valued in $P \times _Kz_0$ , when i is zero it is the space of smooth sections of the associated bundle.
In the two cases of interest to us, we define
Note that in both cases, P admits a left action of $G(\mathbb {R})^+$ and that the associated vector bundles are $G(\mathbb {R})^+$ -equivariant. Moreover, it is a Euclidean bundle, equipped with the inner product
on the fiber. Let $ \Omega ^i(P,z_0)$ be the space of $z_0$ -valued differential i-forms on P. A differential form $\alpha $ in $\Omega ^i(P,z_0)$ is said to be horizontal if $\iota _X\alpha $ vanishes for all vertical vector fields X. There is a left action of K on a differential form $\alpha $ in $\Omega ^i(P,z_0)$ defined by
and $\alpha $ is K-invariant if it satisfies $k\cdot \alpha = \alpha $ for any k in $K,$ i.e., we have $R_k^\ast \alpha = \rho (k^{-1}) \alpha $ . We write $\Omega ^i(P, z_0)^K$ for the space of K-invariant $z_0$ -valued forms on P. Finally, a form that is horizontal and K-invariant is called a basic form and the space of such forms is denoted by $\Omega ^i(P,z_0)_{\mathrm{bas}}$ .
Let $X_1, \dots , X_N$ be tangent vectors of $P/K$ at $\pi (p),$ and let $\widetilde {X}_i$ be tangent vectors of P at p that satisfy $d_p\pi (\widetilde {X}_i)=X_i$ . There is a map
defined by
Proposition 3.1 The map is well-defined and yields an isomorphism between $\Omega ^i(P/K,P \times _K z_0)$ and $\Omega ^i(P,z_0)_{\mathrm{bas}}$ . In particular, if $z_0$ is one-dimensional, then $\Omega ^i(P/K)$ is isomorphic to $\Omega ^i(P)_{\mathrm{bas}}$ .
Proof In the case where i is zero, the horizontally condition is vacuous and the isomorphism simply identifies $\Omega ^0(P/K,P \times _K z_0)$ with $\Omega ^0(P,z_0)^K$ . We have a map
which is well defined since
Conversely, every smooth section s in $\Omega ^0(P/K,P \times _K z_0)$ is given by
for some smooth function $f_s$ in $\Omega ^0(P,z_0)^K$ . The map sending s to $f_s$ is inverse to the previous one. The proof is similar for positive i.
3.3 Covariant derivatives
A covariant derivative on the vector bundle $P \times _K z_0$ is a differential operator
such that for every smooth function f in $C^\infty (P/K),$ we have
The inner product on $P \times _Kz_0$ defines a pairing
and we say that the derivative is compatible with the metric if
for any two sections $s_1$ and $s_2$ in $\Omega ^0(P/K,P\times _Kz_0)$ . There is a covariant derivative that is induced by a principal connection $\theta _P$ in $\Omega ^1(P) \otimes \mathfrak {k}$ as follows. The derivative of the representation gives a map
which we also denote by $\rho $ by abuse of notation. Note that for the representation (3.11), this is simply the map
since $\mathfrak {k}$ splits as $\mathfrak {so}(z_0^\perp ) \oplus \mathfrak {so}(z_0)$ . Composing the principal connection with $\rho $ defines an element
In particular, if s is a section of $P \times _K z_0$ , then we can identify it with a K-invariant smooth map $f_s$ in $\Omega ^0(P, z_0)^K$ . Since $\rho (\theta _P)$ is a $\mathfrak {so}(z_0)$ -valued form and $\mathfrak {so}(z_0)$ is a subspace of $\operatorname {End}(z_0),$ we can define
Lemma 3.2 The form $df_s+\rho (\theta _P) \cdot f_s$ is basic, hence gives a $P \times _K z_0$ -valued form on $P/K$ . Thus, $d+\rho (\theta _P)$ defines a covariant derivative on $P \times _K z_0$ . Moreover, it is compatible with the metric.
Proof See [Reference Berline, Getzler and Vergne1, p. 24]. For the compatibility with the metric, it follows from the fact that the connection $\rho (\theta _P)$ is valued in $\mathfrak {so}(z_0)$ that
Hence, if we denote by $\nabla _P$ is the covariant derivative defined by $d+\rho (\theta _P),$ then
Let us denote by $\nabla _P$ the covariant derivative $d+\rho (\theta _P)$ . It can be extended to a map
by setting
where
We define the curvature $R_P$ in $\Omega ^2(P,\mathfrak {k})$ by
for two vector fields X and Y on P. It is basic by [Reference Berline, Getzler and Vergne1, Proposition 1.13] and composing with $\rho $ gives an element
so that we can view it as an element in $\Omega ^2(P/K,P \times _K \mathfrak {so}(z_0)),$ where K acts on $\mathfrak {so}(z_0)$ by the $\operatorname {Ad}$ -representation. For a section s in $\Omega ^0(P/K,P \times _K z_0)$ , we have [Reference Berline, Getzler and Vergne1, Proposition 1.15]
From now on, we denote by $\nabla $ and $\widetilde {\nabla }$ the covariant derivatives on E and $\widetilde {E}$ associated with $\theta $ and $\widetilde {\theta }$ defined in (3.7) and (3.10). Let R and $\widetilde {R}$ be their respective curvatures.
3.4 Pullback of bundles
The pullback of E by the projection map gives a canonical bundle
over E. We have the following diagram:
The projection induces a pullback of the sections
We can also pullback the covariant derivative $\nabla $ to a covariant derivative
on $\pi ^\ast E$ . It is characterized by the property
Proposition 3.3 The bundles $\widetilde {E}$ and $\pi ^\ast E$ are isomorphic, and this isomorphism identifies $\widetilde {\nabla }$ and $\pi ^\ast \nabla $ .
Proof By definition, $([g_1,w_1],[g_2,w_2])$ are elements of $ \pi ^\ast E$ if and only if $g^{-1}_1g_2$ is in K. We have a $G(\mathbb {R})^+$ -equivariant morphism
This map is well defined and has as inverse
The second statement follows from the fact that $\widetilde {\theta }$ is $\pi ^\ast \theta $ .
3.5 A few operations on the vector bundles
We extend the K-representation $z_0$ to $\bigwedge ^j z_0$ by
We consider the bundles $P \times _K \wedge ^j z_0$ and $ P \times _K \wedge z_0$ over $P/K$ , where $\bigwedge z_0$ is defined as $ \bigoplus _i \bigwedge ^iz_0$ . Denote the space of differential forms valued in $P \times _K \wedge ^j z_0$ by
The total space of differential forms
is an (associative) bigraded $C^\infty (P/K)$ -algebra, where the product is defined by
This algebra structure allows us to define an exponential map by
where $\omega ^k$ is the k-fold wedge product $\omega \wedge \cdots \wedge \omega $ .
Remark 3.1 Suppose that $\omega $ and $\eta $ commute. Then the binomial formula
holds and one can show that $\exp (\omega +\eta )=\exp (\omega )+\exp (\eta )$ in the same way as for the real exponential map. In particular, the diagonal subalgebra $\bigoplus \Omega _P^{i,i}$ is a commutative, since for two forms $\omega $ and $\eta $ in $\Omega _P$ , we have
and similarly for two sections s and t in $\Omega ^0(P/K,P \times _K z_0)$ .
The inner product $\langle - , - \rangle $ on $z_0$ can be extended to an inner product on $\bigwedge z_0$ by
If $e_1, \dots , e_q$ is an orthonormal basis of $z_0$ , then the set
is an orthonormal basis of $\bigwedge z_0$ . We define the Berezin integral $\int ^B$ to be the orthogonal projection onto the top dimensional component, that is the map
The Berezin integral can then be extended to
where $\int ^B s$ in $C^\infty (P/K)$ is the composition of the section with the Berezinian in every fiber. Let $s_1, \dots , s_q$ be a local orthonormal frame of $P \times _K z_0$ . Then $s_1 \wedge \cdots \wedge s_q$ is in $\Omega ^0(P/K,\wedge ^q P \times _K z_0)$ and defines a global section. Hence, for $\alpha $ in $\Omega (P/K,P\times _K \wedge z_0),$ we have
Finally, for every section s in $\Omega ^{0,1}$ , we can define the contraction
and extended by linearity, where the symbol $ \, \widehat {\cdot } \,$ means that we remove it from the product. Note that when j is zero, then $i(s)$ is defined to be zero. The contraction $i(s)$ defines a derivation on $\oplus \widetilde {\Omega }^{i,j}$ that satisfies
for $\alpha $ in $\widetilde {\Omega }^{i,j}$ and $\alpha '$ in $\widetilde {\Omega }^{k,l}$ .
3.6 Thom forms
We denote by E the bundle $G(\mathbb {R})^+\times _K z_0$ . On the fibers of the bundle, we have the inner product given by . Let v be arbitrary vector in L and $\Gamma _v$ its stabilizer. Since the bundle is $G(\mathbb {R})^+$ -equivariant, we have a bundle
and let $\operatorname {D}(\Gamma _v \backslash E)$ be the closed disk bundle. If we have a closed $(q+i)$ -form on $\Gamma _v \backslash E$ whose support is contained in $\operatorname {D}(\Gamma _v \backslash E)$ , then it has compact support in the fiber and represents a class in $H^{q+i}(\Gamma _v \backslash E,\Gamma _v \backslash E-\operatorname {D}(\Gamma _v \backslash E))$ . The cohomology group $H^{\bullet }(\Gamma _v \backslash E,\Gamma _v \backslash E-\operatorname {D}(\Gamma _v \backslash E))$ is equal to the cohomology group $H^{\bullet }(\Gamma _v \backslash E,\Gamma _v \backslash (E-E_0))$ that we used in the introduction, where $E_0$ is the zero section. Fiber integration induces an isomorphism on the level of cohomology
known as the Thom isomorphism [Reference Bott and Tu2, Theorem 6.17]. When i is zero, then $H^i(\Gamma _v \backslash \mathbb {D}^+)$ is $\mathbb {R}$ and we call the preimage of $1$
the Thom class. Any differential form representating this class is called a Thom form, in particular, every closed q-form on $\Gamma _v \backslash E$ that has compact support in every fiber and whose integral along every fiber is $1$ is a Thom form. One can also view the Thom class as the Poincaré dual class of the zero section $E_0$ in E, in the same sense as for (2.24).
Let $\omega $ in $\Omega ^j(E)$ be a form on the bundle, and let $\omega _z$ be its restriction to a fiber $E_z=\pi ^{-1}(z)$ for some z in $\mathbb {D}^+$ . After identifying $z_0$ with $\mathbb {R}^q$ , we see $\omega _z$ as an element of $C^\infty (\mathbb {R}^q) \otimes \wedge ^j(\mathbb {R}^q)^\ast $ . We say that $\omega $ is rapidly decreasing in the fiber, if $\omega _z$ lies in $\mathscr {S}(\mathbb {R}^q) \otimes \wedge ^j(\mathbb {R}^q)^\ast $ for every z in $\mathbb {D}^+$ . We write $\Omega ^j_{\textrm {rd}}(E)$ for the space of such forms.
Let $\Omega ^\bullet _{\textrm {rd}}(\Gamma _v \backslash E)$ be the complex of rapidly decreasing forms in the fiber. It is isomorphic to the complex $\Omega ^\bullet _{\textrm {rd}}(E)^{\Gamma _v}$ of rapidly decreasing $\Gamma _v$ -invariant forms on E. Let $H_{\textrm {rd}}(\Gamma _v \backslash E)$ the cohomology of this complex. The map
is a diffeomorphism from the open disk bundle $\operatorname {D}(\Gamma _v \backslash E)^\circ $ onto $\Gamma _v \backslash E$ . It induces an isomorphism by pullback
which commutes with the fiber integration. Hence, we have the following version of the Thom isomorphism:
The construction of Mathai and Quillen produces a Thom form
which is $G(\mathbb {R})^+$ -invariant (hence, $\Gamma _v$ -invariant) and closed. We will recall their construction in the next section.
3.7 The Mathai–Quillen construction
As earlier, let $\widetilde {E}$ be the bundle $(G(\mathbb {R})^+ \times z_0) \times _K z_0$ . Let $ \wedge ^j \tilde {E} $ be the bundle $(G(\mathbb {R})^+ \times z_0) \times _K \wedge ^j z_0$ and
First, consider the tautological section $\mathbf{s}$ of $\widetilde {E}$ defined by
This gives a canonical element $\mathbf{s}$ of $\widetilde {\Omega }^{0,1}$ . Composing with the norm induced from the inner product, we get an element $\lVert \mathbf{s} \rVert ^2$ in $\widetilde {\Omega }^{0,0}$ .
The representation $\rho $ on $z_0$ induces a representation on $\wedge ^iz_0$ that we also denote by $\rho $ . The derivative at the identity gives a map
The connection form $\rho (\widetilde {\theta })$ in $\Omega ^1(G(\mathbb {R})^+ \times z_0,\wedge ^j z_0)$ defines a covariant derivative
on $\wedge ^j \widetilde {E}$ . We can extend it to a map
by setting
as in (3.30). The connection on $\widetilde {\Omega }^{i,j}$ is compatible with the metric. Finally, the covariant derivative $\widetilde {\nabla }$ defines a derivation on $\oplus \widetilde {\Omega }^{i,j}$ that satisfies
for any $\alpha $ in $\widetilde {\Omega }^{i,j}$ and $\alpha '$ in $\widetilde {\Omega }^{k,l}$ .
Taking the derivative of the tautological section gives an element
Let $\mathfrak {so}(\widetilde {E})$ denote the bundle $(G(\mathbb {R})^+ \times z_0) \times _K \mathfrak {so}(z_0)$ and consider the curvature $\rho (\widetilde {R})$ in $\Omega ^2(\widetilde {E}, \mathfrak {so}(\widetilde {E}))$ . We have an isomorphism
The inverse sends $v \wedge w$ to the endomorphism $ u \mapsto \langle v,u \rangle w-\langle w,u \rangle v$ , and is the isomorphism from (2.11) restricted to $z_0$ . Note that we have
Using this isomorphism, we can also identify $\mathfrak {so}(\widetilde {E})$ and $\wedge ^2 \widetilde {E}$ so that we can view the curvature as an element
Lemma 3.4 The form lying in $\widetilde {\Omega }^{0,0} \oplus \widetilde {\Omega }^{1,1} \oplus \widetilde {\Omega }^{2,2}$ is annihilated by $\widetilde {\nabla }+ 2 \sqrt {\pi } i(\mathbf{s})$ . Moreover
for every form $\alpha $ in $\widetilde {\Omega }^{i,j}$ . Hence, $\int ^B exp(-\omega )$ is a closed form.
Proof We have
It vanishes, because we have the following:
-
∙ $i(\mathbf{s}) \lVert \mathbf{s} \rVert ^2=0$ since $\lVert \mathbf{s} \rVert $ is in $\widetilde {\Omega }^{0,0}$ ,
-
∙ $\widetilde {\nabla } \rho (\widetilde {R})=0$ by Bianchi’s identity,
-
∙ $\widetilde {\nabla }\lVert \mathbf{s} \rVert ^2= 2\langle \widetilde {\nabla }\mathbf{s},\mathbf{s} \rangle =-2 i(\mathbf{s}) \widetilde {\nabla }\mathbf{s} $ ,
-
∙ $\widetilde {\nabla }^2\mathbf{s}=\rho (\widetilde {R})\mathbf{s} =i(\mathbf{s})\rho (\widetilde {R})$ .
For the last point, we used (3.73), where we view $\rho (\widetilde {R})$ as an element of $\Omega ^2(E,\mathfrak {so}(\widetilde {E}))$ , respectively of $\Omega ^2(E,\wedge ^2\widetilde {E})$ .
Let $s_1 \wedge \cdots \wedge s_q$ in $\Omega ^0(E,\wedge ^q \widetilde {E})$ be a global section, where $s_1, \dots , s_q$ is a local orthonormal frame for $\widetilde {E}$ . Then, for any $\alpha $ in $\widetilde {\Omega }^{i,j}$ , we have
This vanishes if j is different from q, hence we can assume $\alpha $ is in $\widetilde {\Omega }^{i,q}$ . If we write $\alpha $ as $\beta s_1 \wedge \cdots \wedge s_q$ for some $\beta $ in $\Omega ^i(E)$ , then
On the other hand, since the connection on $\widetilde {\Omega }^{i,q}$ is compatible with the metric, we have
Then we have
Since $\widetilde {\nabla } + 2 \sqrt {\pi }i(\mathbf{s})$ is a derivation that annihilates $\omega $ , we have
for positive k. Hence, it follows that
In [Reference Mathai and Quillen10], Mathai and Quillen define the following form:
We call it the Mathai–Quillen form.
Proposition 3.5 The Mathai–Quillen form is a Thom form.
Proof From the previous lemma, it follows that the form is closed. It remains to show that its integral along the fibers is $1$ . The restriction of the form $U_{MQ}$ along the fiber $\pi ^{-1}(eK)$ is given by
and its integral over the fiber $\pi ^{-1}(eK)$ is equal to $1$ .
4 Computation of the Mathai–Quillen form
4.1 The section $s_v$
Let $\textrm {pr}$ denote the orthogonal projection of $V(\mathbb {R})$ on the plane $z_0$ . Consider the section
where $g_z$ is any element of $G(\mathbb {R})^+$ sending $z_0$ to z. Let us denote by $L_g$ the left action of an element g in $G(\mathbb {R})^+$ on $\mathbb {D}^+$ . We also denote by $L_g$ the action on E given by $L_{g}[g_z,v]=[gg_z,v]$ . The bundle is $G(\mathbb {R})^+$ -equivariant with respect to these actions.
Proposition 4.1 The section $s_v$ is well-defined and $\Gamma _v$ -equivariant. Moreover, its zero locus is precisely $\mathbb {D}^+_v$ .
Proof The section is well-defined, since replacing $g_z$ by $g_zk$ gives
Suppose that z is in the zero locus of $s_v$ , that is to say $\textrm {pr}(g_z^{-1}v)$ vanishes. Then $g_z^{-1}v$ is in $z_0^\perp $ . It is equivalent to the fact that $z=g_zz_0$ is a subspace of $v^\perp $ , which means that z is in $\mathbb {D}_v^+$ . Hence, the zero locus of $s_v$ is exactly $\mathbb {D}^+_v$ . For the equivariance, note that we have
Hence, if $\gamma $ is an element of $\Gamma _v$ , we have
We define the pullback of the Mathai–Quillen form by $s_v$ . It defines a form
It is only rapidly decreasing on $\mathbb {R}^q$ , and in order to make it rapidly decreasing everywhere we set
It defines a form $ \varphi \in \mathscr {S}(\mathbb {R}^{p+q}) \otimes \Omega ^q(\mathbb {D})^+$ .
Proposition 4.2
-
(1) For fixed v in $V(\mathbb {R}),$ the form $\varphi ^0(v)$ in $\Omega ^q(\mathbb {D}^+)$ is given by
(4.7) $$ \begin{align} \varphi^0(v) & = (- 1)^{\frac{q(q+1)}{2}} (2\pi)^{-\frac{q}{2}} \exp \left (2 \pi {Q\big|{}_{z_0} }(v,v) \right ) \int^B \exp \left (-2 \sqrt{\pi} \nabla s_v+\rho(R)\right ). \end{align} $$ -
(2) It satisfies $L_g^\ast \varphi ^0(v)=\varphi ^0(g^{-1}v)$ , hence
(4.8) $$ \begin{align} \varphi^0 \in \left [\Omega^q(\mathbb{D}^+) \otimes C^\infty(\mathbb{R}^{p+q}) \right ]^{G(\mathbb{R})^+}.\end{align} $$ -
(3) It is a Poincaré dual of $\Gamma _v \backslash \mathbb {D}_v^+$ in $\Gamma _v \backslash \mathbb {D}^+$ .
Proof
-
(1) Recall that $\widetilde {\nabla }=\pi ^\ast \nabla $ and $\widetilde {R}=\pi ^\ast R$ . We pullback by $s_v$
Since $\pi \circ s_v$ is the identity, we have
(4.9) $$ \begin{align} s_v^\ast \widetilde{\nabla} = s_v^\ast \pi^\ast \nabla = \nabla. \end{align} $$Hence, the pullback connection $s_v^\ast \widetilde {\nabla }$ satisfies
(4.10) $$ \begin{align} s_v^\ast (\widetilde{\nabla} \mathbf{s})= (s_v^\ast \widetilde{\nabla}) ( s_v^\ast \mathbf{s}) = \nabla s_v, \end{align} $$since $s_v^\ast \mathbf{s}=s_v$ . We also have $s_v^\ast \widetilde {R}=R$ and(4.11) $$ \begin{align}s_v^\ast \lVert \mathbf{s} \rVert^2= \lVert s_v \rVert^2= \langle s_v , s_v \rangle=-{Q\big|{}_{z_0} }(v,v). \end{align} $$The expression for $\varphi ^0$ then follows from the fact that $\exp $ and $s_v^\ast $ commute.
-
(2) The bundle E is $G(\mathbb {R})^+$ equivariant. By construction, the Mathai–Quillen form is $G(\mathbb {R})^+$ -invariant, so $L_g^\ast U_{MQ}=U_{MQ}$ . On the other hand, we also have
(4.12) $$ \begin{align}s_v \circ L_g(z)=L_g \circ s_{g^{-1}v}(z),\end{align} $$and thus,
(4.13) $$ \begin{align} L_g^\ast \varphi^0(v)=L_g^\ast s_v^\ast U_{MQ}=\varphi^0(g^{-1}v). \end{align} $$ -
(3) Since $s_v$ is $\Gamma _v$ -equivariant, we view it as a section
(4.14) $$ \begin{align} s_v \colon \Gamma_v \backslash \mathbb{D}^+ \longrightarrow \Gamma_v \backslash E, \end{align} $$whose zero locus is precisely $\Gamma _v \backslash \mathbb {D}_v^+$ . Let $S_0$ (resp. $S_v$ ) be the image in $\Gamma _v \backslash E$ of the section $s_v$ (resp. the zero section). By [Reference Bott and Tu2, Proposition 6.24(b)], the Thom form $U_{MQ}$ is a Poincaré dual of the zero section $S_0$ of E. For a form $\omega $ in $\Omega _c^{m-q}(\Gamma _v \backslash \mathbb {D}^+),$ we have(4.15) $$ \begin{align} \int_{\Gamma_v \backslash \mathbb{D}^+} \varphi^0(v) \wedge \omega & = \int_{\Gamma_v \backslash \mathbb{D}^+} s_v^\ast \left ( U_{MQ} \wedge \pi ^\ast \omega \right ) \nonumber \\ & = \int_{S_v} U_{MQ} \wedge \pi ^\ast \omega \nonumber \\ & = \int_{S_v \cap S_0} \pi ^\ast \omega \nonumber \\ & = \int_{\Gamma_v \backslash \mathbb{D}_v^+} \omega. \end{align} $$The last step follows from the fact that $\pi ^{-1}(S_v \cap S_0)$ equals $\Gamma _v \backslash \mathbb {D}_v^+$ .
As in (2.19), we have an isomorphism
by evaluating at the basepoint $eK$ of $G(\mathbb {R})^+/K$ that corresponds to $z_0$ in $\mathbb {D}^+$ . We will now compute ${ {\varphi ^0}\big|{}_{eK} }$ .
4.2 The Mathai–Quillen form at the identity
From now on, we identify $\mathbb {R}^{p+q}$ with $V(\mathbb {R})$ by the orthonormal basis of (2.1), and let $z_0$ be the negative spanned by the vectors $e_{p+1}, \ldots , e_{p+q}$ . Hence, we identify $z_0$ with $\mathbb {R}^q$ and the quadratic form is
where $x_{p+1}, \dots , x_{p+q}$ are the coordinates of the vector v.
Let $f_v$ in $\Omega ^0(G(\mathbb {R})^+,z_0)^K$ be the map associated with the section $s_v$ , as in Proposition 3.1. It is defined by
Then $df_v+\rho (\theta ) f_v$ is the horizontal lift of $\nabla s_v$ , as discussed in Section 3.1. Let X be a vector in $\mathfrak {g}$ , and let $X_{\mathfrak {p}}$ and $X_{\mathfrak {k}}$ be its components with respect to the splitting of $\mathfrak {g}$ as $\mathfrak {p} \oplus \mathfrak {k}$ . We have
In particular, we can evaluate on the basis $X_{\alpha \mu }$ and get:
So as an element of $\mathfrak {p}^\ast \otimes z_0$ , we can write
with
Proposition 4.3 Let $\rho (R_e)$ in $\wedge ^2\mathfrak {p}^\ast \otimes \mathfrak {so}(z_0)$ be the curvature at the identity. Then after identifying $\mathfrak {so}(z_0)$ with $\wedge ^2 z_0$ , we have
where $\eta _\alpha ^2=\eta _\alpha \wedge \eta _\alpha $ .
Proof Using the relation $E_{ij}E_{kl}=\delta _{il}E_{kj}$ , one can show that
for two vectors $X_{\alpha \nu }$ and $ X_{\beta \mu }$ in $\mathfrak {p}$ . Hence, we have
On the other hand, since $\eta _i(X_{jr})=\delta _{ij}e_r$ , we also have
The lemma follows since $\rho (X_{\nu \mu })=T(e_\nu \wedge e_\mu )$ in $\mathfrak {so}(z_0)$ , because
Using the fact that the exponential satisfies $\exp (\omega +\eta )=\exp (\omega )\exp (\eta )$ on the subalgebra $\bigoplus \Omega ^{i,i}$ —see Remark 3.1—we can write
We define the nth Hermite polynomial by
The first three Hermite polynomials are $H_0(x)=1$ , $H_1(x)=2x$ , and $H_2(x)=4x^2-2$ .
Lemma 4.4 Let $\eta $ be a form in $\bigoplus \Omega ^{i,i}$ . Then
where $H_n$ is the nth Hermite polynomial.
Proof Since $\eta $ and $\eta ^2$ are in $\bigoplus \Omega ^{i,i}$ , they commute and we can use the binomial formula:
where
The conditions on k and l imply that n is less than or equal to $2k$ . First, suppose that n is even. Then we have that k is between $\frac {n}{2}$ and n, so that the sum above can be written
where in the second step, we let m be $k-\frac {n}{2}$ . If n is odd, then k is between $\frac {n+1}{2}$ and n, so that the sum can be written
Applying the lemma to (4.28), we get
If $n_1+ \cdots + n_p$ is different from q, then the Berezinian of $\eta _1^{n_1} \wedge \cdots \wedge \eta _p^{n_p}$ vanishes and we get
Note that
where the sums are over all $\mu _i$ ’s between $p+1$ and $p+q$ . If $n_1+\cdots +n_p$ is equal to $q,$ we have
where the sums in the last two lines go over all tuples $\underline {\alpha }=(\alpha _1, \dots , \alpha _q)$ with $\alpha $ between $1$ and p, and the value $\alpha $ appears exactly $n_{\alpha }$ -times in $\underline {\alpha }$ . Hence
After multiplying by $\exp \left (-\pi Q(v,v) \right )$ , we get
The form is now rapidly decreasing in v, since the Siegel majorant is positive definite. We have
Theorem 4.5 We have $2^{-\frac {q}{2}}\varphi (v)=\varphi _{KM}(v)$ .
Proof It is a straightforward computation to show that
Hence, applying this, we find that the Kudla–Millson form, defined by the Howe operators in (2.22), is
5 Examples and remarks
-
(1) Let us compute the Kudla–Millson as above in the simplest setting of signature $(1,1)$ . Let $V(\mathbb {R})$ be the quadratic space $\mathbb {R}^2$ with the quadratic form $Q(v,w)=x'y+xy'$ , where x and $x'$ (resp. y and $y'$ ) are the components of v (respectively of w). Let $e_1=\frac {1}{\sqrt {2}}(1,1)$ and $e_2=\frac {1}{\sqrt {2}}(1,-1)$ . The one-dimensional negative plane $z_0$ is $\mathbb {R} e_2$ . If r denotes the variable on $z_0$ , then the quadratic form is ${{Q}\big|{}_{z_0} }(r)=-r^2$ . The projection map is given by
(5.1) $$ \begin{align} \operatorname{pr} \colon V(\mathbb{R}) & \longrightarrow z_0 \nonumber \\ v=(x,x') & \longmapsto \frac{x-x'}{\sqrt{2}}. \end{align} $$The orthogonal group of $V(\mathbb {R})$ is
(5.2) $$ \begin{align} G(\mathbb{R})^+=\left \{ \begin{pmatrix} t & 0 \\ 0 & t^{-1} \end{pmatrix},t>0 \right \}, \end{align} $$and $\mathbb {D}^+$ can be identified with $\mathbb {R}_{>0}$ . The associated bundle E is $\mathbb {R}_{>0} \times \mathbb {R}$ and the connection $\nabla $ is simply d since the bundle is trivial. Hence, the Mathai–Quillen form is(5.3) $$ \begin{align} U_{MQ}=\sqrt{2}e^{-2\pi r^2}dr \in \Omega^1(E), \end{align} $$as in the proof of Proposition 3.5. The section $s_v \colon \mathbb {R}_{>0} \rightarrow E$ is given by(5.4) $$ \begin{align} s_v(t)=\left (t, \frac{t^{-1}x-tx'}{\sqrt{2}} \right ), \end{align} $$where x and $x'$ are the components of v. We obtain(5.5) $$ \begin{align} s_v^\ast U_{MQ}=e^{-\pi \left ( \frac{x}{t}-tx' \right )^2}\left ( \frac{x}{t}+tx' \right ) \frac{dt}{t}. \end{align} $$Hence, after multiplication by $2^{-\frac {1}{2}}e^{-\pi Q(v,v)}$ , we get
(5.6) $$ \begin{align} \varphi_{KM}(x,x')= 2^{-\frac{1}{2}}e^{-\pi \left [ \left (\frac{x}{t} \right )^2+(tx')^2 \right ]}\left ( \frac{x}{t}+tx' \right ) \frac{dt}{t}. \end{align} $$ -
(2) The second example illustrates the functorial properties of the Mathai–Quillen form. Suppose that we have an orthogonal splitting of $V(\mathbb {R})$ as $\bigoplus _i^r V_i(\mathbb {R})$ . Let $(p_i,q_i)$ be the signature of $V_i(\mathbb {R})$ . We have
(5.7) $$ \begin{align} \mathbb{D}_{1} \times \cdots \times \mathbb{D}_{r} \simeq \left \{ z \in \mathbb{D} \; \vert \; z = \bigoplus_{i=1}^r z \cap V_i(\mathbb{R}) \right \}. \end{align} $$Suppose, we fix $z_0= z_0^1 \oplus \cdots \oplus z_0^r$ in $\mathbb {D}^+_{1} \times \cdots \times \mathbb {D}^+_{r} \subset \mathbb {D}$ , where $z_0^i$ is a negative $q_i$ -plane in $V_i(\mathbb {R})$ . Let $G_i(\mathbb {R})$ be the subgroup preserving $V_i(\mathbb {R})$ , let $K_i$ be the stabilizer of $z_0^i$ , and $\mathbb {D}_i$ be the symmetric space associated with $V_i(\mathbb {R})$ .
Over $\mathbb {D}^+_{1} \times \cdots \times \mathbb {D}^+_{r}$ the bundle E splits as an orthogonal sum $E_1 \oplus \cdots \oplus E_r$ , where $E_i$ is the bundle $G_i(\mathbb {R})^+ \times _{K_i} z_0^i$ . Moreover, the restriction of the Mathai–Quillen form to this subbundle is
(5.8) $$ \begin{align} {{U_{MQ}}\big|{}_{E_1 \times \cdots \times E_r} }=U_{MQ}^1 \wedge \cdots \wedge U_{MQ}^r, \end{align} $$where $U_{MQ}^i$ is the Mathai–Quillen form on $E_i$ . The section $s_v$ also splits as a direct sum $\oplus s_{v_i}$ , where $v_i$ is the projection of v onto $v_i$ . In summary, the following diagram commutes(5.9)and we can conclude that(5.10) $$ \begin{align} {{\varphi_{KM}(v)}\big|{}_{\mathbb{D}_1^+ \times \cdots \times \mathbb{D}_r^+} }=\varphi_{KM}^1(v_1)\wedge \cdots \wedge \varphi_{KM}^r(v_r), \end{align} $$where $\varphi _{KM}^i$ is the Kudla–Millson form on $\mathbb {D}_i^+$ . -
(2) Let $U \subset V$ be a nondegenerate r-subspace spanned by vectors $v_1, \dots , v_r$ . Let $(p',q')$ be the signature of U. Let $\mathbb {D}_U$ be the subspace
(5.11)When U is positive, i.e., when $q'=0$ , then $\mathbb {D}_U$ is in fact
(5.12)In particular, when U is spanned by a single positive vector v, then $\mathbb {D}_U=\mathbb {D}_v$ , where $\mathbb {D}_v$ is as in (2.4). Kudla and Millson construct an $rq$ -form $\varphi _{KM}(v_1,\dots ,v_r)$ that is a Poincaré dual to $\Gamma _U \backslash \mathbb {D}_U$ in $\Gamma _U \backslash \mathbb {D}$ , where $\Gamma _U$ is the stabilizer of U in $\Gamma $ . One of its properties [Reference Kudla and Millson8][Lemma. 4.1] is that
(5.13) $$ \begin{align} \varphi_{KM}(v_1,\dots,v_r)=\varphi_{KM}(v_1) \wedge \cdots \wedge \varphi_{KM}(v_r). \end{align} $$Let us explain how this form can also be recovered by the Mathai–Quillen formalism. Consider the bundle $E^r=E \oplus \cdots \oplus E$ of rank $rq$ over $\mathbb {D}$ . One can check that all the “ingredients” of the Mathai–Quillen form $U_{MQ}(E^r)$ are compatible with respect to the splitting as a direct sum, so that we have
(5.14) $$ \begin{align} U_{MQ}(E^r)=U_{MQ}(E) \wedge \dots \wedge U_{MQ}(E). \end{align} $$On the other hand, the zero locus of the section of $E^r$ is precisely $\mathbb {D}_U$ . Hence, the pullback
(5.15)is a Poincaré dual of $\mathbb {D}_U$ . Moreover, by (5.14), we have(5.16) $$ \begin{align} \varphi^0(v_1,\dots,v_r)=\varphi^0(v_1) \wedge \cdots \wedge \varphi^0(v_r). \end{align} $$Finally, after setting
(5.17)we get(5.18) $$ \begin{align} 2^{-\frac{rq}{2}}\varphi(v_1,\dots, v_r) & = 2^{-\frac{rq}{2}} e^{-\pi \sum_{i=1}^r Q(v_i,v_i) }\varphi^0(v_1) \wedge \cdots \wedge \varphi^0(v_r) \nonumber\\ & = 2^{-\frac{rq}{2}} \varphi(v_1) \wedge \cdots \wedge \varphi(v_r) \nonumber\\ & = \varphi_{KM}(v_1) \wedge \cdots \wedge \varphi_{KM}(v_r) \nonumber \\ & = \varphi_{KM}(v_1,\dots,v_r). \end{align} $$
Acknowledgment
This project is part of my thesis and I thank my advisors Nicolas Bergeron and Luis Garcia for suggesting me this topic and for their support. I thank the anonymous referee for helpful comments and suggestions.