Nadya Gurevich, Edmund Karasiewicz
We prove the Casselman-Shalika formula for unramified groups over a non-archimedean local field by studying the action of the spherical Hecke algebra on the space of compact spherical Whittaker functions via the twisted Satake transform. This method provides a conceptual explanation of the appearance of characters of a dual group in the Casselman-Shalika formula.
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The Casselman-Shalika method is a way to compute explicit formulas for periods of irreducible unramified representations of p-adic groups that are associated to unique models (i.e. multiplicity-free induced representations). We apply this method to the case of the Shalika model of GL_n, which is known to distinguish lifts from odd orthogonal groups. In the course of our proof, we further develop a variant of the method, that was introduced by Y.Hironaka, and in effect reduce many such problems to straightforward calculations on the group.
This paper studies spherical Whittaker functions for central extensions of reductive groups over local fields. We follow the development of Chinta and Offen to produce a metaplectic Casselman-Shalika formula for tame covers of all unramified groups.
Geometric Casselman-Shalika in mixed characteristic
本文在混合特征局部域上构建了分裂连通半单群的几何卡森-沙利卡公式，构造了Witt向量仿射格拉斯曼ian上的sheaves，计算了其上同调。
For an arbitrary split adjoint group we identify the unramified Whittaker space with the space of skew-invariant functions on the lattice of coweights and deduce from it the Casselman-Shalika formula.
On a Casselman-Shalika type formula for unramified Speh representations
本文给出了非分歧Speh表示的Casselman-Shalika型公式，利用修正Hall-Littlewood多项式计算(k,c)模型的规范化球面元素在特定矩阵处的值，并采用组合方法证明，解答了Lapid-Mao的问题。
In this paper we establish a Steinberg-Lusztig tensor product theorem for abstract Fock space. This is a generalization of the type A result of Leclerc-Thibon and a Grothendieck group version of the Steinberg-Lusztig tensor product theorem for representations of quantum groups at roots of unity. Although the statement can be phrased in terms of parabolic affine Kazhdan-Lusztig polynomials and thus has geometric content, our proof is combinatorial, using the theory of crystals (Littelmann paths). We derive the Casselman-Shalika formula as a consequence of the Steinberg-Lusztig tensor product th…
A Betti geometric Casselman-Shalika equivalence
本文基于贝蒂sheaves研究Kirillov模型，证明了贝蒂几何卡森-沙利卡等价，将仿射Grassmannian上的Iwahori-Whittaker perverse sheaves与Satake范畴联系起来。
In the recent works of Brubaker-Bump-Friedberg, Bump-Nakasuji, and others, the product in the Casselman-Shalika formula is written as a sum over a crystal. The coefficient of each crystal element is defined using the data coming from the whole crystal graph structure. In this paper, we adopt the tableaux model for the crystal and obtain the same coefficients using data from each individual tableaux; i.e., we do not need to look at the graph structure. We also show how to combine our results with tensor products of crystals to obtain the sum of coefficients for a given weight. The sum is a q-po…
A Tale of Two Idempotents: Casselman-Shalika and Spherical Genericity
该研究通过伊瓦霍-赫克代数的球型与符号幂等元，无需非分歧主系列或交错算子，推导了Casselman-Shalika公式与J.-S. Li的非分歧群不可约球型表示的泛性质判据。
We present a novel integral representation for a quotient of global automorphic L-functions, the symmetric square over the exterior square. The pole of this integral characterizes a period of a residual representation of an Eisenstein series. As such, the integral itself constitutes a period, of an arithmetic nature. The construction involves the study of local and global aspects of a new model for double covers of general linear groups, the metaplectic Shalika model. In particular, we prove uniqueness results over p-adic and Archimedean fields, and a new Casselman-Shalika type formula.
For a reductive group $G$ over a non-Archimedean local field (e.g $GL_n( \mathbb{Q}_p )$ ), Jacquet's Whittaker function is essentially proportional to a character of an irreducible representation of the Langlands dual group $G^\vee( \mathbb{C} )$ ( a Schur function if $G = GL_n( \mathbb{Q}_p )$). We propose a probabilistic approach to this claim, known as the Shintani-Casselman-Shalika formula, when the group $G$ has at least one minuscule cocharacter in the coweight lattice. Our presentation goes along the following lines. Thanks to a minuscule random walk $W^{(z)}$ on the coweight lattice a…
We define unramified Whittaker functions on the p-adic points of an affine Kac-Moody group, and establish an analogue of the Casselman-Shalika formula for these functions.
Spherical Shalika models on $\mathrm{PGU}_{2,2}$ and the theta correspondence for $(\mathrm{PGSp}_4,\mathrm{PGU}_{2,2})$
本文研究局部域上PGU_{2,2}的generic非分歧表示的Shalika模型，利用theta对应证明其在常数范围内唯一，并给出将球面Shalika泛函值与PGSp_4对偶群表示相联系的Casselman-Shalika公式。
We obtain explicit formulas for the product of a deformed Weyl denominator with the character of an irreducible representation of the spin group $\rm{Spin}_{2r+1}({\mathbb C})$, which is an analogue of the formulas of Tokuyama for Schur polynomials and Hamel-King for characters of symplectic groups. To give these, we start with a symplectic group and obtain such characters using the Casselman-Shalika formula. We then analyze this using objects which are naturally attached to the metaplectic double cover of an odd orthogonal group, which also has dual group $\rm{Spin}_{2r+1}({\mathbb C})$.
Let X=H\G be a homogeneous spherical variety for a split reductive group G over the integers o of a p-adic field k, and K=G(o) a hyperspecial maximal compact subgroup of G=G(k). We compute eigenfunctions ("spherical functions") on X=X(k) under the action of the unramified (or spherical) Hecke algebra of G, generalizing many classical results of "Casselman-Shalika" type. Under some additional assumptions on X we also prove a variant of the formula which involves a certain quotient of L-values, and we present several applications such as: (1) a statement on "good test vectors" in the multiplicit…
Spherical Whittaker functions on the metaplectic n-fold cover of GL(r+1) over a nonarchimedean local field containing n distinct n-th roots of unity may be expressed as the partition functions of statistical mechanical systems that are variants of the six-vertex model. If n=1 then in view of the Casselman-Shalika formula this fact is related to Tokuyama's deformation of the Weyl character formula. It is shown that various properties of these Whittaker functions may be expressed in terms of the commutativity of row transfer matrices for the system. Potentially these properties (which are alread…
We introduce the (global) q-Whittaker function as the limit at t=0 of the q,t-spherical function extending the symmetric Macdonald polynomials to arbitrary eigenvalues. The construction heavily depends on the technique of the q-Gaussians developed by the author (and Stokman in the non-reduced case). In this approach, the q-Whittaker function is given by a series convergent everywhere, a kind of generating function for multi-dimensional q-Hermite polynomials (closely related to the level 1 Demazure characters). One of the applications is a q-version of the Shintani- Casselman- Shalika formula, …
We prove a `motivic' analogue of the Weyl character formula, computing the Euler characteristic of a line bundle on a generalized flag manifold $G/B$ multiplied either by a motivic Chern class of a Schubert cell, or a Segre analogue of it. The result, given in terms of Demazure-Lusztig (D-L) operators, recovers formulas found by Brubaker, Bump and Licata for the Iwahori-Whittaker functions of the principal series representation of a $p$-adic group. In particular, we obtain a new proof of the classical Casselman-Shalika formula for the spherical Whittaker function. The proofs are based on local…
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