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Dani Szpruch

Publications and source records attributed to Dani Szpruch.

11 recordsLinked to original sources
arXiv2024

Whittaker spaces for reducible unitary principal series representations of $\widetilde{SL_2(F)}$

Let $F$ be a $p$-adic field containing the full group of $n^{th}$ roots of 1 and let $ \widetilde{SL_2(F)}$ be the $n$-fold cover of $SL_2(F)$ constructed by Kubota. In this paper we compute the dimension of the space of Whittaker functionals of the two irreducible summands inside a reducible unitary genuine principal series representation of $\widetilde{SL_2(F)}$. We also show how these dimensions change when the Whittaker character is modified. As an application we determine the action of the twisted Kazhdan-Patterson $n$-fold cover of $GL_2(F)$ on the two summands. We emphasize that our main results addresses both ramified and unramified representations and do not rely on the assumption that the cover is tame.

math.NT↗
arXiv2024

A p-adic analog of Hasse-Davenport product relation involving epsilon-factors

In this paper we prove some generalizations of the classical Hasse-Davenport product relation for certain arithmetic factors defined on p-adic fields, among them one finds the epsilon-factors appearing in Tate's thesis. We then show that these generalizations are equivalent to some representation theoretic identities relating the determinant of ramified local coefficients matrices defined for coverings of SL(2,F) to Plancherel measures and gamma-factors.

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arXiv2021

Restrictions, L-parameters, and local coefficients for genuine representations

We consider the restriction and induction of representations between a covering group and its derived subgroup, both on the representation-theoretic side and the L-parameter side. In particular, restriction of a genuine principal series is analyzed in detail. We also discuss a metaplectic tensor product construction for covers of the symplectic similitudes groups, and remark on the generality of such a construction for other groups. Furthermore, working with an arbitrary irreducible constituent of a unitary unramified principal series, we prove a multiplicity formula for its restriction to the derived subgroup in terms of three associated R-groups. Later in the paper, we study an unramified L-packet on how the parametrization of elements inside such a packet varies along with different choices of hyperspecial maximal compact subgroups and their splittings. We also investigate the genericity of elements inside such an L-packet with respect to varying Whittaker datum. Pertaining to the above two problems, covers of the symplectic similitudes groups are discussed in detail in the last part of the paper.

math.RT↗
arXiv2019

Local coefficients and gamma factors for principal series of covering groups

We consider an $n$-fold Brylinski-Deligne cover of a reductive group over a $p$-adic field. Since the space of Whittaker functionals of an irreducible genuine representation of such a cover is not one-dimensional, one can consider a local coefficients matrix arising from an intertwining operator, which is the natural analogue of the local coefficients in the linear case. In this paper, we concentrate on genuine principal series and establish some fundamental properties of such a local coefficients matrix, including the investigation of its arithmetic invariants. As a consequence, we prove a form of the Casselman-Shalika formula which could be viewed as a natural analogue for linear algebraic groups. We also investigate in some depth the behaviour of the local coefficients matrix with respect to the restriction of genuine principal series from covers of ${\rm GL}_2$ to ${\rm SL}_2$. In particular, some further relations are unveiled between local coefficients matrices and gamma factors or metaplectic-gamma factors.

math.RT↗
arXiv2018

On Shahidi local coefficients matrix

In these notes we define and study the Shahidi local coefficients matrix associated with a genuine principal series representation I(σ) of an n-fold cover of p-adic SL(2,F) and an additive character ψ. The conjugacy class of this matrix is an invariant of the inducing representation σ and ψ and its entries are linear combinations of Tate or Tate type γ-factors. We relate these entries to functional equations associated with linear maps defined on the dual of the space of Schwartz functions. As an application we give new formulas for the Plancherel measure and use these to relate principal series representations of different coverings of SL(2,F). While we do not assume that the residual characteristic of F is relatively prime to n we do assume that n is not divisible by 4.

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arXiv2015

Plancherel measures for coverings of p-adic SL(2,F)

In these notes we compute the Plancherel measures associated with genuine principal series representations of n-fold covers of p-adic SL(2,F). Along the way we also compute a higher dimensional metaplectic analog of Shahidi local coefficients. Our method involves new functional equations utilizing the Tate Gamma-factor and a metaplectic counterpart. As an application we prove an irreducibility theorem.

math.NT↗
arXiv2014

The non-tempered theta 10 Arthur parameter and Gross-Prasad Conjectures

We provide a construction of local and automorphic non-tempered Arthur packets of the group SO(3,2) and its inner form SO(4,1) associated with a certain Arthur's parameter and prove a multiplicity formula. We further study the restriction of the representations in these packets to the subgroup SO(3,1). In particular, we discover that the local Gross-Prasad conjecture, formulated for generic L-packets, does not generalize naively to a non-generic A-packet. We also study the non-vanishing of the automorphic SO(3,1)-period on the group SO(4,1) x SO(3,1) and SO(3,2) x SO(3,1) for the representations above. The main tool is the local and global theta correspondence for unitary quaternionic similitude dual pairs.

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arXiv2013

Some results in the theory of genuine representations of the metaplectic double cover of GSp2n(F) over p-adic fields

Let F be a p-adic field and let G(n) and G`(n) be the metaplectic double covers of the general symplectic group and symplectic group attached to a 2n dimensional symplectic space over F. We show here that if n is odd then all the genuine irreducible representations of G(n) are induced from a normal subgroup of finite index closely related to G`(n). Thus, we reduce, in this case, the theory of genuine admissible representations of G(n) to the better understood corresponding theory of G`(n). For odd n we also prove the uniqueness of certain Whittaker functionals along with Rodier type of Heredity. Our results apply also to all parabolic subgroups of G(n) if n is odd and to some of the parabolic subgroups of G(n) if n is even. We prove some irreducibility criteria for parabolic induction on G(n) for both even and odd n. As a corollary we show, among other results, that while for odd n, all genuine principal series representations of G(n) induced from unitary representations are irreducible, there exist reducibility points on the unitary axis if n is even. We also list all the reducible genuine principal series representations of G(2) provided that the F is not 2-adic.

math.NT↗
arXiv2013

Symmetric genuine Spherical Whittaker functions on the metaplectic double cover of GSp(2n,F)

Let F be a p-adic field of odd residual characteristic. Let G(n) and G`(n) be the metaplectic double covers of the general symplectic group and the symplectic group attached to the 2n dimensional symplectic space over F. Let T be a genuine, possibly reducible, unramified principal series representation of G(n). In these notes we give an explicit formulas for a spanning set for the space of Spherical Whittaker functions attached to T. For odd n, and generically for even n, this spanning set is a basis. The signicant property of this set is that each of its elements is unchanged under the action of the Weyl group of G`(n). If n is odd then each element in the set has an equivariant property that generalizes the uniqueness result of Gelbart, Howe and Piatetski-Shapiro proven for G(1). Using this symmetric set, we construct a family of reducible genuine unramified principal series representations which have more then one generic constituent. This family contains all the reducible genuine unramified principal series representations induced from a unitary data and exists only for n even.

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arXiv2010

The Langlands-Shahidi Method for the metaplectic group and applications

I am applying the Langlands-Shahidi method to the metaplectic double cover of Sp(2n). I proved that a Whittaker model of an irreducible admissible representation of this group is unique. As a result I was able to define the local coefficients for this group. I used them to determine irreducibility of parabolic induction. I also found some connections with the representation theory of SO(2n+1). I have defined local gamma factors and proved some properties of them.

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