Whittaker coefficients of geometric Eisenstein series

Let $G$ be a simply connected complex reductive group with Langlands dual group $\check{G}$ defined over $k=\mathbf{C}$. Choose a maximal torus $T$ and a Borel subgroup $B$ with unipotent radical $N$. Let $\rho$ be half the sum of the positive coroots. Let $X$ be a smooth projective complex genus $g$ curve. Choose a square root of the canonical bundle on $X$ and form the anticanonical $T$-bundle $\omega^{-\rho}$. All categories and functors are derived.

Let $\sigma$ be a $\check{T}$-local system on $X$, and let $\operatorname{Loc}_{\check{N}}^{\sigma}=\operatorname{Loc}_{\check{B}}\times_{\operatorname{Loc}_{\check{T}}}\sigma$ be the derived moduli stack of $\check{B}$-local systems on $X$ whose underlying $\check{T}$-local system is identified with $\sigma$, see (1.3). A $\check{T}$-local system is called regular if for every coroot the associated rank 1 local system is nontrivial. If $\sigma$ is regular then $\operatorname{Loc}_{\check{N}}^{\sigma}$ is a classical affine scheme isomorphic to a vector space.

Let $K$ be the Hecke $\sigma$-eigensheaf on $\operatorname{Bun}_{T}$ whose stalk at $\omega^{-\rho}$ twisted by a negative coweight valued divisor $\underline{\lambda}\cdot\underline{x}=\sum\lambda_{i}x_{i}$ is

Above, $\sigma^{\lambda}_{x}$ means the fiber at $x$ of the rank 1 local system obtained from $\sigma$ using $\lambda$. Here $d_{T}=\dim\operatorname{Bun}_{T}$ and $d^{\lambda}=\langle 2\check{\rho},4(g-1)\rho+\lambda\rangle$ is the shift appearing in section 6.4.8 of [Gai13].

The Whittaker or Poincaré series sheaf $\operatorname{Whit}=r_{!}\chi^{*}D\exp=r_{!}(-\chi)^{*}\exp[-2]$ on $\operatorname{Bun}_{G}$ is the pullback then pushforward of the exponential sheaf along

see 5.4.1 of [FR22]. The function $\chi$ is defined in for example [FGV01]. The character sheaf $\exp$ on $\mathbf{A}^{1}$ is normalized so that its costalks are in degree zero. Up to a shift, its Verdier dual $D\exp$ is the inverse character sheaf. Both $\exp$ and $D\exp$ corepresent shifted vanishing cycles for conic sheaves on $\mathbf{A}^{1}$, so the distinction is not so important. In the Betti setting we do not have the exponential D-module. Because $\chi$ is $\mathbf{C}^{\times}$-equivariant for the $2\rho$ action on $\operatorname{Bun}_{N^{-}}^{\omega^{-\rho}}$ and the weight 2 action on $\mathbf{A}^{1}$, the sheaf defined in 2.5.2 of [NY19] serves as a substitute. The Whittaker sheaf does not have nilpotent singular support.

The automorphic and spectral Eisenstein series functors, $\operatorname{Eis}_{!}=p_{!}q^{*}$ and $\operatorname{\check{E}is}=\check{p}^{\operatorname{IndCoh}}_{*}\check{q}^{\operatorname{IndCoh}*}$, are defined by pullback then pushforward along

All of the above functors are left adjoints, in particular $\check{p}^{\operatorname{IndCoh}}_{*}$ is a left adjoint because $\check{p}$ is proper. In (1.2), the functor $\operatorname{Eis}_{!}$ is modified according to section 4.1 of [Gai10] or section 6.4.8 [Gai13]. This matches the translation by $\omega^{-\rho}$ and shift by $d^{\lambda}$ built into our definition of $K$ in (1.1).

The geometric Langlands conjecture is supposed to be compatible with parabolic induction. Moreover the Whittaker functional is expected to correspond under Langlands to global sections on $\operatorname{Loc}_{\check{G}}$, up to a shift by $d_{G}=\dim\operatorname{Bun}_{G}$. Thus commutativity of conjectural diagram

applied to the skyscraper $k_{\sigma}$, predicts the following isomorphism.

The proof uses a combination of [Ras21] and [BG08] to relate twisted cohomology of the Zastava space to the formal completion of $\operatorname{Loc}_{\check{N}}^{\sigma}$.

Both sides of the main theorem are coweight graded vector spaces. On the automorphic side, let $K^{\lambda}$ be the restriction to the degree $-\lambda-2(g-1)\rho$ connected component $\operatorname{Bun}_{T}^{\lambda}$. On the spectral side, the adjoint $\check{T}$-action on $\check{B}$ induces an action on $\operatorname{Loc}_{\check{N}}^{\sigma}$.

Our results apply for all three versions of geometric Langlands: de Rham, restricted, and Betti. On the automorphic side $\operatorname{Eis}_{!}K$ is a constructible sheaf, equivalently regular holonomic D-module, with nilpotent singular support, see [Gin01]. On the spectral side there are three versions of the moduli space of local systems, all having the same complex valued points. For a unipotent group

coincide by proposition 4.3.3 and section 4.8.1 of [AGK⁺20].

It is convenient to take the coweight graded linear dual to avoid topological rings and because Lie algebra homology behaves better than Lie algebra cohomology. Here is the proof of our main theorem in one sentence:

In section 2.1, we use [NT22] or [FR22] to exchange $\operatorname{Eis}_{!}$ for a right adjoint, then apply base change and a result of [AG15] to get a calculation on the Zastava space. In section 2.2, we pushforward to the space of positive coweight valued divisors and, by theorem 4.6.1 of [Ras21], obtain a certain factorizable perverse sheaves $\Upsilon_{\sigma}^{\lambda}$ on $X^{\lambda}$.

In section 2.3, we interpret $\Upsilon_{\sigma}$ in terms of the chiral enveloping algebra of $\check{\mathfrak{n}}_{\sigma}$ as in [BG08]. In section 2.4, we explain, following [BG08], how the cohomology of $\Upsilon_{\sigma}$ equals factorization homology of $A=C_{\bullet}(\check{\mathfrak{n}}_{\sigma})$. Beilinson and Drinfeld’s formula says factorization homology of $C_{\bullet}(\check{\mathfrak{n}}_{\sigma})$ is Lie algebra homology of $\Gamma(X,\check{\mathfrak{n}}_{\sigma})$. In section 2.5, we study moduli of $\check{\mathfrak{n}}$-local systems using deformation theory. Since $\Gamma(X,\check{\mathfrak{n}}_{\sigma})$ is the shifted tangent complex of $\operatorname{Loc}_{\check{N}}^{\sigma}$, its Lie algebra homology is related the formal completion of $\operatorname{Loc}_{\check{N}}^{\sigma}$ at $\sigma$. Using that $\operatorname{Loc}_{\check{N}}^{\sigma}=(\operatorname{Spec}R)/\check{N}$ is the quotient of an affine scheme by a unipotent group and using the contracting $\mathbf{G}_{m}$-action, we show that $C_{\bullet}(\Gamma(X,\check{\mathfrak{n}}_{\sigma}))=\mathcal{O}(\operatorname{Loc}_{\check{N}}^{\sigma})^{*}$ is the graded linear dual ring of functions.

The idea of using factorization homology to study the formal completion of $\operatorname{Loc}_{\check{N}}^{\sigma}$ is from [BG08]. For $\sigma$ regular, propositions 11.3 and 11.4 of [BG08] give an isomorphism between $\prod\Gamma(X^{\lambda},\Upsilon_{\sigma}^{\lambda})^{*}$ and the completed ring of functions $\mathcal{O}(\operatorname{Loc}_{\check{N}}^{\sigma})^{\wedge}$. Sections 2.3 and 2.4 review some of their arguments and do not contain new content apart from filling in some details. Our main contribution is in section 2.5 where we extend the results of [BG08] to the more interesting case of irregular $\sigma$, and obtain a formula for the ring of functions on $\operatorname{Loc}_{\check{N}}^{\sigma}$ (not just its formal completion) using the contracting $\mathbf{G}_{m}$-action.

I thank David Nadler for suggesting Whittaker coefficients of Eisenstein series and for generous discussions. This work was partially supported by NSF grant DMS-1646385.

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

The fiber product $Z^{\prime}=\operatorname{Bun}_{B}\times_{\operatorname{Bun}_{G}}\operatorname{Bun}_{N^{-}}^{\omega^{-\rho}}$ has a stratification indexed by the Weyl group, determined by the generic relative position of two flags. Let $j:Z\hookrightarrow Z^{\prime}$ be the open inclusion of the locus where the two flags are generically transverse, called the Zastava space.

Consider the compositions

and let $q_{Z}=q_{Z^{\prime}}j$ and $\chi_{Z}=\chi_{Z^{\prime}}j$ be their restrictions to $Z$.

In this section we recall how to factor the projection $q_{Z}:Z^{\lambda}\rightarrow\operatorname{Bun}_{T}^{\lambda}$ through the configuration space $X^{\lambda}$ of positive coweight valued divisors of total degree $\lambda$. Hence a description of the $\lambda$-graded piece of proposition 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 $E$ with generically transverse $B,B^{-}$-reductions $F,F^{-}$, such that $F$ has degree $-\lambda-2(g-1)\rho$ and $F^{-}\times_{B^{-}}T=\omega^{-\rho}$. For each dominant weight $\check{\mu}$ the Plucker description gives maps

Here $F^{\check{\mu}}=F\times_{B}\mathbf{C}_{\check{\mu}}$ is a line bundle and $E^{\check{\mu}}=E\times_{G}V_{\check{\mu}}$ 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 $F^{\check{\mu}}(\langle\underline{x}\cdot\underline{\lambda},\check{\mu}\rangle)=\omega^{-\langle\check{\mu},\rho\rangle}$. Since $G$ is assumed simply connected, we can write $\lambda=\sum n_{i}\alpha_{i}$ as a sum of simple coroots and $X^{\lambda}=\prod X^{(n_{i})}$ is 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,

Let $\lambda$ be a coweight and $n=\langle\check{\rho},\lambda\rangle$. Let $\check{\mathfrak{n}}_{\sigma}=\sigma\times_{\check{T}}\check{\mathfrak{n}}$, an $\check{\mathfrak{n}}$-local system on $X$. The Chevalley complex on the coweight graded Ran space gives a $\prod S_{n_{i}}$ equivariant sheaf on $\prod X^{n_{i}}$. Symmetrizing by pushing forward along $\prod X^{n_{i}}/S_{n_{i}}\rightarrow X^{\lambda}$ gives a perverse sheaf $\Upsilon_{\sigma}^{\lambda}$, see section 4 of [Ras21] and equation (2.6). 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 [BG08]. The stalk of $\Upsilon_{\sigma}^{\lambda}$ at a positive coweight valued divisor $\underline{x}\cdot\underline{\lambda}\in X^{\lambda}$ is

Combining propositions 1 and 2 shows Whittaker coefficients of Eisenstein series is graded dual to global sections of $\Upsilon_{\sigma}$ on the configuration space.

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

A sheaf on the Ran space of $X$ is a collection of sheaves $A_{X^{I}}$ on each power of the curve $X^{I}$ compatible under $!$-restriction along all partial diagonal maps, see 4.2.1 of [BD04] or 2.1 of [FG12]. Recall from section 1.2.1 of [FG12] that the category of sheaves on the Ran space admits two tensor products with a map $\otimes^{*}\rightarrow\otimes^{\operatorname{ch}}$ between them.

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

Let $A:=C_{\bullet}(\check{\mathfrak{n}}_{\sigma})\in\operatorname{Shv}(\operatorname{Ran})$ 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 $\operatorname{ch}$-tensor product. Proposition 6.1.2 of [FG12] 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 $\operatorname{Sym}(\check{\mathfrak{n}}_{\sigma}[1])$ is coweight graded, and because the Chevalley differential preserves the grading. Choose a coweight $\lambda$ and let $n=\langle\check{\rho},\lambda\rangle$. The sheaf $A^{\lambda}_{X^{n}}$ on $X^{n}$ is $S_{n}$-equivariant and perverse. Symmetrize it along $\operatorname{sym}:X^{n}\rightarrow X^{(n)}$ to get a perverse sheaf $(\operatorname{sym}_{*}A^{\lambda}_{X^{n}})^{S_{n}}$ on the $n$th 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 perverse subsheaf $A^{\lambda}_{X^{(n)}}\subset(\operatorname{sym}_{*}A^{\lambda}_{X^{n}})^{S_{n}}$ defined in section 3 of [BG08]. Let $X_{i}^{(n)}\subset X^{(n)}$ be the space of effective degree $n$ divisors supported at exactly $i$ points. The $!$-restriction of $(\operatorname{sym}_{*}A^{\lambda}_{X^{n}})^{S_{n}}$ 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

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

By section 11.6 of [BG08], the pushforward of $\Upsilon_{\sigma}^{\lambda}$, see (2.3), along the partial symmetrization map $\operatorname{sym}^{\lambda}:X^{\lambda}\rightarrow X^{(n)}$ is

In this section we review, following [BG08], how factorization homology of $A^{\lambda}=C_{\bullet}(\check{\mathfrak{n}}_{\sigma})^{\lambda}$ can be computed as cohomology on the symmetric power $X^{(n)}$, where $n=\langle\check{\rho},\lambda\rangle$.

Let $\operatorname{FSet}$ 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 $\Delta:X^{I}\rightarrow X^{J}$. By definition of a sheaf on the Ran space $A_{X^{I}}=\Delta^{!}A_{X^{J}}$ so adjunction gives maps $\Delta_{*}A_{X^{I}}\rightarrow A_{X^{J}}$. Factorization homology is defined in section 6.3.3 of [FG12] or section 4.2.2 of [BD04] as the colimit over these maps

The following proposition is stated in 11.6 of [BG08] and below we fill in the proof using the Cousin filtration and ideas from section 4.2 of [BD04].

Since the factorization algebra $A=C_{\bullet}(\check{\mathfrak{n}}_{\sigma})$ 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 [BD04] or 6.4.4 of [FG12], says

In this section we show that

Lie algebra homology of the shifted tangent complex equals the graded dual ring of functions on $\operatorname{Loc}_{\check{N}}^{\sigma}$. Deformation theory says that $C_{\bullet}(\Gamma(X,\check{\mathfrak{n}}_{\sigma}))=\Gamma^{\operatorname{IndCoh}}(\omega_{(\operatorname{Loc}_{\check{N}}^{\sigma})^{\wedge}})$ is global sections of the dualizing sheaf on the formal completion at $\sigma$. Using the structure of $\operatorname{Loc}_{\check{N}}^{\sigma}$ described in proposition 4, we can recover the graded dual ring of functions on $\operatorname{Loc}_{\check{N}}^{\sigma}$, not just its completion, from $\Gamma^{\operatorname{IndCoh}}(\omega_{(\operatorname{Loc}_{\check{N}}^{\sigma})^{\wedge}})$.

First we show that $\operatorname{Loc}_{\check{N}}^{\sigma}=\operatorname{Loc}_{\check{N}}^{\sigma,x}/\check{N}$ is the quotient by a unipotent group of an affine derived scheme with a contracting $\mathbf{G}_{m}$-action. Let $\operatorname{Loc}_{\check{B}}^{x}=\check{B}^{2g}\times_{\check{B}}1$ (respectively $\operatorname{Loc}_{\check{T}}^{x}=\check{T}^{2g}\times_{\check{T}}1$) be the Betti moduli of $\check{B}$ (respectively $\check{T}$) local systems trivialized at a point $x$. Let $\operatorname{Loc}_{\check{N}}^{\sigma,x}=\operatorname{Loc}_{\check{B}}^{x}\times_{\operatorname{Loc}_{\check{T}}^{x}}\sigma$ 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 $\sigma\in\operatorname{Loc}_{\check{T}}$ so there is a canonical lift $\sigma\in\operatorname{Loc}_{\check{T}}^{x}$. We also sometimes regard $\sigma$ as a point in $\operatorname{Loc}_{\check{N}}^{\sigma,x}$ via the inclusion $\check{T}\subset\check{B}$.

Let $\check{B}$ act on $\operatorname{Loc}_{\check{B}}^{x}$ 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 $\operatorname{Loc}_{\check{B}}^{x}$ to $\operatorname{Loc}_{\check{T}}^{x}$, as is made precise below.

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 [GR17b] or remark 2.4.2 of [Lur11], there is an equivalence

between Lie algebra modules for the shifted tangent complex and indcoherent sheaves on the formal completion. Let $p:Y^{\wedge}\rightarrow\operatorname{pt}$ be the map to a point. By chapter 7 section 5.2 of [GR17b], the trivial $T_{\sigma}Y[-1]$-module corresponds to the dualizing sheaf $\omega_{Y^{\wedge}}=p^{!}k\in\operatorname{IndCoh}(Y^{\wedge})$. Moreover Lie algebra homology corresponds to global sections

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

Suppose $Y^{\wedge}=\operatorname{Spf}R^{\wedge}=\operatorname{colim}Y_{n}$ where $Y_{n}=\operatorname{Spec}R/\mathfrak{m}^{n}$ and let $i_{n}:Y_{n}\rightarrow Y^{\wedge}$. Since $Y_{n}\rightarrow Y_{n+1}$ is proper, $\operatorname{IndCoh}(Y^{\wedge})$ is the colimit under $*$-pushforward of $\operatorname{IndCoh}(Y_{n})$, see chapter 1 proposition 2.5.7 of [GR17a]. The dualizing sheaf can be written as a colimit, $\omega_{Y^{\wedge}}=\operatorname{colim}i_{n*}^{\operatorname{IndCoh}}\omega_{Y_{n}}$, see chapter 7 corollary 5.3.3 of [GR17b]. Since $\Gamma^{\operatorname{IndCoh}}(Y^{\wedge},-)$ is continuous it follows that

is the topological dual of the completed local ring $R^{\wedge}$. In this case, equation (2.13) is corollary 5.2 of [Hin96].

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