Geometric Casselman–Shalika Equivalence

• Geometric Casselman–Shalika equivalence is a categorical framework that reinterprets spherical Hecke modules using Whittaker sheaves with inherent Weyl skew symmetry.
• It employs the affine Grassmannian to bridge classical Whittaker models with dual-group representation theory and integrates tools like Jacquet functors and parabolic restrictions.
• The framework adapts across various settings—étale, mixed-characteristic, Betti, and motivic—providing insights for the geometric Langlands program and beyond.

Searching arXiv for recent and foundational papers on geometric Casselman–Shalika. Geometric Casselman–Shalika equivalence is the geometric and categorical form of the principle that Whittaker objects on the affine Grassmannian encode the same dual-group representation theory that geometric Satake assigns to spherical objects, but with the Whittaker condition inserting the Weyl-sign, or skew, symmetry that classically underlies the Casselman–Shalika formula. In its decategorified form, this principle identifies spherical Hecke data with Weyl-invariant functions and unramified Whittaker data with Weyl-skew-invariant functions on the coweight lattice, from which the classical character formula follows; in its geometric form, it appears as equivalences between Whittaker or Iwahori–Whittaker categories and Satake-type categories, together with compatibilities for Jacquet, Eisenstein, and constant term functors (Gurevich, 2013, Faergeman et al., 18 Jul 2025, Iyengar et al., 2024, Sandvik, 1 Apr 2026, Cass et al., 24 Mar 2026).

1. Algebraic prototype: Satake, Whittaker, and Weyl skew-symmetry

For a split adjoint group $G$ over a non-archimedean local field, the classical spherical Hecke algebra

$H_K=C_c(K\backslash G/K)$

is identified by the Satake transform with the $W$-invariant part of the coweight-lattice algebra, while the unramified Whittaker space $(\mathrm{Ind}_N^G\psi)^K$ is identified with the $W$-skew-invariant part

$$\mathbb C[A]^{W,-}=\{f\in \mathbb C[A]:w(f)=(-1)^{\ell(w)}f\}.$$

The central algebraic statement is the canonical $H_K$-module isomorphism

$$j:(\mathrm{Ind}_N^G\psi)^K\xrightarrow{\sim}\mathbb C[A]^{W,-},$$

compatible with

$H_K\simeq \mathbb C[A]^W,$

and sending the Whittaker basis vector $\phi_{\lambda+\rho}$ to the alternating sum $H_K=C_c(K\backslash G/K)$0 (Gurevich, 2013).

This formulation isolates the structural core of Casselman–Shalika. The spherical side is governed by Weyl invariants; the Whittaker side is governed by the sign-isotypic part for the Weyl action. The usual formula

$H_K=C_c(K\backslash G/K)$1

then becomes a consequence of module compatibility rather than a separate harmonic-analytic computation (Gurevich, 2013).

For connected unramified quasi-split groups, the same mechanism persists through a twisted Satake transform on Iwahori-level data. In that setting, the kernel is explicitly generated by skew-symmetry relations

$H_K=C_c(K\backslash G/K)$2

and the compact spherical Whittaker space is identified with the alternating part of $H_K=C_c(K\backslash G/K)$3. This yields a conceptual explanation of why dual-group characters appear in the Casselman–Shalika formula: alternating elements factor through the Weyl denominator, so the Whittaker recursion is transported representation-ring multiplication on the dual side (Gurevich et al., 2020).

2. From decategorification to geometry on the affine Grassmannian

The geometric version replaces functions and Hecke modules by sheaf-theoretic categories on the affine Grassmannian. In the factorization formulation, the geometric Casselman–Shalika equivalence is an equivalence of factorization module categories

$H_K=C_c(K\backslash G/K)$4

whose fiber at a point $H_K=C_c(K\backslash G/K)$5 gives the local equivalence

$H_K=C_c(K\backslash G/K)$6

Under this equivalence, the Whittaker generator $H_K=C_c(K\backslash G/K)$7 is sent to the irreducible $H_K=C_c(K\backslash G/K)$8-representation of highest weight $H_K=C_c(K\backslash G/K)$9 (Faergeman et al., 18 Jul 2025).

The significance of this statement is not merely that two abelian or derived categories are abstractly equivalent. The equivalence is formulated so as to respect factorization and Hecke actions. That compatibility is the categorical analogue of the algebraic compatibility between $W$0 and the skew module $W$1. A plausible implication is that the sign representation visible in the function-theoretic Whittaker model is realized geometrically through the semi-infinite and Whittaker structures built into the affine-Grassmannian category.

This geometric form explains the usual slogan that Casselman–Shalika is a shadow of geometric Satake. Geometric Satake identifies spherical sheaves with representations of $W$2; the Casselman–Shalika refinement identifies Whittaker sheaves with the same representation theory, but through a category in which the Whittaker condition has already absorbed the Weyl alternation.

3. Jacquet functors, parabolic restriction, and constant terms

A major refinement of the theory is the parabolic compatibility theorem. For a parabolic $W$3 with Levi $W$4, the natural Jacquet functors on Whittaker categories are shown to match the expected representation-theoretic restriction procedures under geometric Casselman–Shalika. More precisely, $W$5 corresponds to ordinary restriction

$W$6

whereas $W$7 corresponds to Lie algebra cohomology along the nilradical,

$W$8

The result is expressed as commutativity of two diagrams intertwining $W$9, $(\mathrm{Ind}_N^G\psi)^K$0, the Jacquet functors, and the dual-side restriction or cohomology functors (Faergeman et al., 18 Jul 2025).

The proof uses Hecke structures, Drinfeld–Plücker structures, and enhanced Drinfeld–Plücker structures on kernels of geometric Eisenstein functors. The local semi-infinite IC sheaf

$(\mathrm{Ind}_N^G\psi)^K$1

is characterized by Hecke-theoretic induction and carries a coaction of $(\mathrm{Ind}_N^G\psi)^K$2. Two distinguished semi-infinite objects, $(\mathrm{Ind}_N^G\psi)^K$3 and $(\mathrm{Ind}_N^G\psi)^K$4, produce the functors $(\mathrm{Ind}_N^G\psi)^K$5 and $(\mathrm{Ind}_N^G\psi)^K$6, respectively. The bridge between them is a Koszul-duality statement,

$(\mathrm{Ind}_N^G\psi)^K$7

which converts the compactified object into the open one by passing to invariants (Faergeman et al., 18 Jul 2025).

This parabolic form has immediate relevance to geometric Langlands. The paper presents it as the local ingredient needed to show that the spectral-to-automorphic geometric Langlands functor commutes with constant term functors. In that sense, geometric Casselman–Shalika is not only a local equivalence between Whittaker sheaves and representations; it is also a functorial compatibility principle governing Eisenstein and constant term constructions.

4. Mixed-characteristic geometric Casselman–Shalika

In mixed characteristic, the role of the affine Grassmannian is played by the Witt vector affine Grassmannian $(\mathrm{Ind}_N^G\psi)^K$8, whose $(\mathrm{Ind}_N^G\psi)^K$9-points are $W$0. The geometric input consists of Schubert closures $W$1, their IC sheaves $W$2, the semi-infinite orbits $W$3, and the intersections

$W$4

which serve as mixed-characteristic MV intersections (Iyengar et al., 2024).

The Whittaker datum is geometrized by morphisms

$W$5

and a rank-one local system $W$6 on finite-level quotients $W$7. The need for this construction is specific to mixed characteristic: there is no residue map $W$8 analogous to equal characteristic, so the additive character must be realized through finite Witt-vector quotients rather than an Artin–Schreier sheaf (Iyengar et al., 2024).

The main theorem computes the compactly supported cohomology of

$W$9

on $$\mathbb C[A]^{W,-}=\{f\in \mathbb C[A]:w(f)=(-1)^{\ell(w)}f\}.$$0. For dominant $$\mathbb C[A]^{W,-}=\{f\in \mathbb C[A]:w(f)=(-1)^{\ell(w)}f\}.$$1, the cohomology is concentrated in degree $$\mathbb C[A]^{W,-}=\{f\in \mathbb C[A]:w(f)=(-1)^{\ell(w)}f\}.$$2, and it is one-dimensional exactly in the diagonal case $$\mathbb C[A]^{W,-}=\{f\in \mathbb C[A]:w(f)=(-1)^{\ell(w)}f\}.$$3; it vanishes otherwise (Iyengar et al., 2024). Via the sheaf-function dictionary, this recovers the classical identity

$$\mathbb C[A]^{W,-}=\{f\in \mathbb C[A]:w(f)=(-1)^{\ell(w)}f\}.$$4

The importance of this result is methodological as well as formal. The proof requires auxiliary torsors, convolution decompositions, and a detailed analysis of quasi-minuscule orbits because mixed characteristic lacks a clean Birkhoff decomposition and a negative loop group with the same formal properties as in equal characteristic (Iyengar et al., 2024).

5. Betti and motivic forms of the equivalence

A distinct problem arises in the Betti setting: Whittaker sheaves are usually defined using an exponential $$\mathbb C[A]^{W,-}=\{f\in \mathbb C[A]:w(f)=(-1)^{\ell(w)}f\}.$$5-module or an Artin–Schreier sheaf, but no analogous nontrivial multiplicative local system exists on $$\mathbb C[A]^{W,-}=\{f\in \mathbb C[A]:w(f)=(-1)^{\ell(w)}f\}.$$6 for Betti sheaves. The solution is the Kirillov model of Gaitsgory–Lysenko, implemented through the Fourier–Laumon kernels $$\mathbb C[A]^{W,-}=\{f\in \mathbb C[A]:w(f)=(-1)^{\ell(w)}f\}.$$7 and $$\mathbb C[A]^{W,-}=\{f\in \mathbb C[A]:w(f)=(-1)^{\ell(w)}f\}.$$8. Using this framework, one obtains categories of Betti Iwahori–Whittaker sheaves on the affine Grassmannian and translation functors

$$\mathbb C[A]^{W,-}=\{f\in \mathbb C[A]:w(f)=(-1)^{\ell(w)}f\}.$$9

which are inverse equivalences and $H_K$0-exact (Sandvik, 1 Apr 2026).

The main Betti theorem is an equivalence

$H_K$1

realized by convolution with the lowest standard Whittaker object: $H_K$2 The functor is perverse $H_K$3-exact and restricts to an equivalence between the spherical Satake category and the Iwahori–Whittaker perverse category (Sandvik, 1 Apr 2026). This is a genuine Betti geometric Casselman–Shalika equivalence.

The motivic version goes further by replacing sheaves with motives and Whittaker categories with exponential motives. For a prestack $H_K$4 with a $H_K$5-action, the exponential category is

$H_K$6

and on the affine Grassmannian the resulting Casselman–Shalika equivalence takes the form

$H_K$7

with reduced version

$H_K$8

Its decategorification yields a Whittaker module for the spherical Hecke algebra that is free of rank $H_K$9, and after specialization to a finite field and a choice of additive character it recovers the classical Whittaker module (Cass et al., 24 Mar 2026).

Taken together, these results show that geometric Casselman–Shalika is not tied to a single sheaf theory. Étale, mixed-characteristic, Betti, and motivic realizations all preserve the same organizing principle: the Whittaker side is a categorical model of the spherical Hecke module whose decategorification is the Casselman–Shalika formula.

6. Variants, analogues, and conceptual boundaries

The expression “geometric Casselman–Shalika equivalence” does not denote a single theorem with universally fixed hypotheses. Some works prove categorical equivalences on the affine Grassmannian; others prove compatibilities with parabolic Jacquet functors; others provide algebraic, combinatorial, or representation-theoretic shadows of the same phenomenon.

A prominent combinatorial shadow is the derivation of Casselman–Shalika from the Steinberg–Lusztig tensor product theorem in abstract Fock space. In that setting, the key identity

$$j:(\mathrm{Ind}_N^G\psi)^K\xrightarrow{\sim}\mathbb C[A]^{W,-},$$0

is obtained through canonical bases, affine Hecke algebra realizations, and crystal combinatorics via Littelmann paths. The paper is explicit that it does not claim a new geometric Casselman–Shalika theorem in the direct Braverman–Gaitsgory or Frenkel–Gaitsgory–Vilonen sense; rather, it proves an equivalence of formulations in which the geometric content is encoded by parabolic affine Kazhdan–Lusztig polynomials and affine Hecke algebra geometry (Lanini et al., 2018).

There are also deformations and analogues outside the ordinary spherical Whittaker setting. The metaplectic Casselman–Shalika formula for tame covers of unramified reductive groups replaces the ordinary Weyl-character picture by a metaplectic Weyl-group action and links spherical Whittaker functions to Weyl-group multiple Dirichlet series; the paper states that it does not prove a geometric Casselman–Shalika theorem, but it suggests such an interpretation (McNamara, 2011). For $$j:(\mathrm{Ind}_N^G\psi)^K\xrightarrow{\sim}\mathbb C[A]^{W,-},$$1, a Casselman–Shalika-type formula for spherical Shalika models is derived using theta correspondence with $$j:(\mathrm{Ind}_N^G\psi)^K\xrightarrow{\sim}\mathbb C[A]^{W,-},$$2, so that the relevant dual-group representation theory is that of $$j:(\mathrm{Ind}_N^G\psi)^K\xrightarrow{\sim}\mathbb C[A]^{W,-},$$3 rather than the original unitary group (Cauchi et al., 2023).

Finally, some recent Satake-type results for Kac–Moody groups are described as “very close” to a geometric Casselman–Shalika story. In the affine Kac–Moody setting, the normalized constant term functor is $$j:(\mathrm{Ind}_N^G\psi)^K\xrightarrow{\sim}\mathbb C[A]^{W,-},$$4-exact, IC complexes match irreducible highest-weight representations of the dual group, and affine MV-cycle geometry recovers weight multiplicities. This suggests that the Casselman–Shalika paradigm extends beyond reductive groups whenever constant term geometry and semi-infinite orbit theory retain enough control (Bouthier et al., 13 Oct 2025).

In this broader perspective, geometric Casselman–Shalika equivalence is best viewed as a family of statements centered on one invariant pattern: spherical Hecke theory corresponds to dual-group representation theory, Whittaker conditions impose Weyl alternation, and the classical Casselman–Shalika formula is the decategorified character identity arising from that correspondence.