Klingen $${mathfrak {p}}^2$$ p 2 vectors for $$mathrm{GSp}(4)$$ GSp ( 4 )

The Ramanujan Journal - We calculate the dimensions of the spaces of invariant vectors under the Klingen congruence subgroup of level $${mathfrak {p}}^2$$ for all irreducible, admissible...     

Critical values of L-functions on GSp 2 × Gl 2

This paper deals with Deligne's conjecture on the critical values of L-functions. Let Z _{G ⊗ h} (s) denote the tensor product L-function attached to a Siegel modular form G of weight k and an elliptic cusp form h of weight l. We assume that the first Fourier-Jacobi coefficient of G is not identically zero. Then Deligne's conjecture is fully proven for Z _{G ⊗ h} (s), when l≤2k−2 and partly for the remaining case. Supported by the von Neumann Fund at the Institute for Advanced Study, Princeton     

Parahoric Restriction for GSp(4)

Parahoric restriction is the parahoric analogue of Jacquet’s functor. The group GSp(4, F) of symplectic similitudes of genus two over a local number field F/? _p has five conjugacy classes of parahoric subgroups. For each we determine the parahoric restriction of the non-cuspidal irreducible smooth representations of GSp(4, F) in terms of explicit character values.     

Minuscule representations and Panyushev conjectures

Recently, Panyushev (2015) raised five conjectures concerning the structure of certain root posets arising from Z-gradings of simple Lie algebras. This paper aims to provide proofs for four of them. Our study also links these posets with Kostant-Macdonald identity, minuscule representations, Stembridge’s “t = ?1 phenomenon”, and the cyclic sieving phenomenon due to Reiner et al. (2004).     

Locally Quadratic Modules and Minuscule Representations

We give a new, geometric proof of a theorem by Timmesfeld showing that for simple Chevalley groups, abstract modules where all roots act quadratically are direct sums of minuscule representations. Our proof is uniform, treats finite and infinite fields on an equal footing, and includes Lie rings.     

A Kuznetsov Formula for Kloosterman Sums on GLn

We prove a Kuznetsov sum formula for Kloosterman sums on GL _n corresponding to the big Bruhat cell. Using this formula, a weighted sum of Kloosterman sums can be expressed in spectral decomposition. A non-trivial estimate of the spectral side might lead to a proof of cancellations in sums of Kloosterman sums on GL _n.     

Whittaker functions for generalized principal series representations of GSp(2,R)

We study Whittaker functions for generalized principal series representations of $\mathrm{GSp}(2,\mathbf{R})$ induced from Siegel parabolic subgroup. We give Mellin–Barnes type integral representations of Whittaker functions belonging to certain K-types. As an application of our explicit formulas, we compute the archimedean parts of Novodvorsky's zeta integrals. As a consequence we can show the entireness of the spinor L-functions for generic cusp forms on $\mathrm{GSp}(2)$.     

(GLn,GLm)-duality and symmetric plethysm

In [7] the author has given an exposition of the theory of invariants of binary forms in terms of a particular version of Classical Invariant Theory. Reflection shows that many aspects of the development apply also ton-ary forms. The purpose of this paper is to make explicit this more general application. The plethysms S’(S^p(ℂⁿ)) are computed quite explicitly forl = 2, 3 and 4. Partially supported by NSF Grant No. DMS 8506130.     

The fundamental lemma for the Bessel and Novodvorsky subgroups of GSp(4)

We announce, in the case of the group GSp(4), an equality of two local integrals. One is a Kloosterman integral on the Bessel subgroups of GSp(4) and the other is a Kloosterman integral on the Novodvorsky subgroups of GSp(4). We conjecture that Jacquet's relative trace formula for GL(2) in [7], where Jacquet has given another proof of Waldspurger's result [9], generalizes to GSp(4). We believe that this approach will lead us to a proof and also a precise formulation of a conjecture of Böcherer [1]. Support for this conjecture may be found in the important paper of Böcherer and Schulze-Pillot [2]. Our result serves as the fundamental lemma for our conjectural relative trace formula for the main relevant double cosets.     

A remark on the failure of multiplicity one for GSp(4)

We revisit a classical result of Howe and Piatetski-Shapiro on the failure of strong multiplicity one for GSp(4).     

Minuscule representations and Panyushev conjectures Dedicated to the Memory of Professor Bertram Kostant

Recently, Panyushev(2015) raised five conjectures concerning the structure of certain root posets arising from Z-gradings of simple Lie algebras. This paper aims to provide proofs for four of them. Our study also links these posets with Kostant-Macdonald identity, minuscule representations, Stembridge's t =-1 phenomenon, and the cyclic sieving phenomenon due to Reiner et al.(2004).     

Euler Tours of Maximum Girth in K2 n +1 and K2 n ,2 n

Given an eulerian graph G and an Euler tour T of G, the girth of T, denoted by g(T), is the minimum integer k such that some segment of k+1 consecutive vertices of T is a cycle of length k in G. Let g^E(G)= maxg(T) where the maximum is taken over all Euler tours of G.We prove that g^E(K_2n,2n)=4n–4 and 2n–3g^E(K_2n+1)2n–1 for any n2. We also show that g^E(K₇)=4. We use these results to prove the following:1)The graph K_2n,2n can be decomposed into edge disjoint paths of length k if and only if k4n–1 and the number of edges in K_2n,2n is divisible by k.2)The graph K_2n+1 can be decomposed into edge disjoint paths of length k if and only if k2n and the number edges in K_2n+1 is divisible by k.     

Symmetric solutions for (n,n−2) games with small values ofv (n−2)

Symmetric solutions and their uniqueness are investigated for symmetric games in which onlyn?2 andn players enter into problem. As a result, we present two types of symmetric solutions for the case of smallv (n?2), and a unique one for much smallerv (n-2).