Siegel Vectors for Nongeneric Depth Zero Supercuspidals of GSp4 2411 04973v1

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Source: Cohen, Jonathan. “Siegel Vectors for Nongeneric Depth Zero Supercuspidals of GSp (4).” Preprint, arXiv:2411.04973v1 (2024).
Main Theme: This paper investigates a specific dimension counting problem in the representation theory of the group GSp(4) over non-archimedean local fields of characteristic zero. It focuses on calculating the dimensions of spaces of fixed vectors (Siegel vectors) for a particular class of representations called "depth zero, irreducible, nongeneric supercuspidal representations".
Key Ideas and Facts:
Motivation: The motivation stems from the challenge of computing the dimension of spaces of cusp forms in the classical theory of Siegel modular forms. Understanding the dimensions of spaces of fixed vectors in p-adic representations is crucial for tackling this problem.
Representations Considered: The paper focuses specifically on depth zero supercuspidal representations of GSp(4) that arise from the maximal compact subgroup K(p). These representations can be constructed via compact induction from irreducible representations of certain subgroups.
Approach: The core of the paper lies in reducing the dimension computation to the determination and enumeration of specific double cosets and performing character table computations for finite reductive groups.
Main Results:Dimension Formula: Theorem 5.1 provides an explicit formula for calculating the dimension of the space of Siegel vectors. This formula differentiates between odd and even residue field characteristics (q).
Atkin-Lehner Signature: Theorem 6.2 computes the signature of the Atkin-Lehner involution acting on the space of Siegel vectors. This signature reveals a dichotomy between even and odd n (level of the Siegel congruence subgroup).
Dichotomy: A striking observation is the stark difference in behavior between the cases of even and odd residue field characteristics. This difference highlights the complexity and richness of the problem.
Siegel vs. Klingen Vectors: Unlike higher depth Klingen subgroups, where the considered representations have no fixed vectors, the theory of Siegel vectors exhibits greater depth and complexity.
Applications: The results presented in the paper have direct implications for the study of Siegel modular forms and related areas, particularly in determining the dimensions of spaces of cusp forms.
Summary
This academic paper, written by Jonathan Cohen, calculates the dimensions of spaces of Siegel-invariant vectors in depth-zero, irreducible, nongeneric supercuspidal representations of the symplectic group GSp(4, F), where F is a non-archimedean local field of characteristic zero. The paper establishes a formula for these dimensions and also computes the signature of the Atkin-Lehner involution on these spaces. The motivation for this research comes from the classical theory of Siegel modular forms, where similar dimension-counting problems are encountered. The paper also highlights interesting distinctions in the results between the cases where the residual characteristic p is equal to 2 and p is greater than 2.
Glossary
GSp(4, F): The group of 4x4 symplectic similitudes over the field F.
Depth zero supercuspidal representation: A specific type of irreducible smooth representation of a p-adic group, characterized by its depth being zero and its supercuspidality.
Siegel congruence subgroup: A subgroup of GSp(4,F) defined by specific congruence conditions modulo powers of the prime p.
Siegel vector: A vector in a representation that is invariant under the action of a Siegel congruence subgroup.
K(p): A specific maximal compact subgroup of GSp(4,F) used in the construction of supercuspidal representations.
Double coset: A set of elements of the form HgK, where H and K are subgroups of a group G, and g is an element of G.
Rg: A subgroup of K(p)/K(p)+ associated with a double coset representative g, crucial for calculating dimensions of spaces of fixed vectors.
Character table: A table summarizing the values of the characters of a group's irreducible representations, used to compute dimensions of fixed vector spaces.
Atkin-Lehner involution: An operator acting on representations, often associated with a specific element of the group, used to study the structure of the representation space.
Signature: In the context of the Atkin-Lehner involution, it refers to the difference between the dimensions of the eigenspaces corresponding to the eigenvalues +1 and -1.
Nongeneric representation: A representation that does not possess a Whittaker model, leading to specific properties and behavior in the context of the paper's analysis.
Cuspidal representation: An irreducible representation of a reductive group whose matrix coefficients have specific vanishing properties.