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Steven Spallone

Publications and source records attributed to Steven Spallone.

29 records · Page 2Linked to original sources
arXiv2016

Representations of symmetric groups with non-trivial determinant

We give a closed formula for the number of partitions $λ$ of $n$ such that the corresponding irreducible representation $V_λ$ of $S_n$ has non-trivial determinant. We determine how many of these partitions are self-conjugate and how many are hooks. This is achieved by characterizing the $2$-core towers of such partitions. We also obtain a formula for the number of partitions of $n$ such that the associated permutation representation of $S_n$ has non-trivial determinant.

math.RT↗
arXiv2013

Similarity of matrices over local rings of length two

Let $R$ be a local principal ideal ring of length two, for example, the ring $R=\Z/p^2\Z$ with $p$ prime. In this paper we develop a theory of normal forms for similarity classes in the matrix rings $M_n(R)$ by interpreting them in terms of extensions of $R[t]$-modules. Using this theory, we describe the similarity classes in $M_n(R)$ for $n\leq 4$, along with their centralizers. Among these, we characterize those classes which are similar to their transposes. Non-self-transpose classes are shown to exist for all $n>3$. When $R$ has finite residue field of order $q$, we enumerate the similarity classes and the cardinalities of their centralizers as polynomials in $q$. Surprisingly, the polynomials representing the number of similarity classes in $M_n(R)$ turn out to have non-negative integer coefficients.

math.RA↗
arXiv2013

An integration formula for unipotent radicals

Let P be a maximal parabolic of a classical group over a field F. Then the Levi subgroup M is isomorphic to the product of a classical group and a general linear group, acting on vector spaces X and W, respectively. In this paper we decompose the unipotent radical N of P under the adjoint action of M, assuming dim W is less than or equal to dim X and that dim W is even. When F is a local field, we obtain a Weyl-type integration formula for N.

math.RT↗
arXiv2012

Signs, involutions and Jacquet modules

Let $G$ be a connected reductive $p$-adic group and let $θ$ be an automorphism of $G$ of order at most two. Suppose $π$ is an irreducible smooth representation of $G$ that is taken to its dual by $θ$. The space $V$ of $π$ then carries a non-zero bilinear form $(\mspace{7mu},\mspace{6mu})$, unique up to scaling, with the invariance property $(π(g)v, π({}^θg)w) = (v,w)$, for $g \in G$ and $v, w \in V$. The form is easily seen to be symmetric or skew-symmetric and we set $\varepsilon_θ(π) = \pm1$ accordingly. We use Cassleman's pairing (in commonly observed circumstances) to express $\varepsilon_θ(π)$ in terms of certain Jacquet modules of $π$ and thus, via the Langlands classification, reduce the problem of determining the sign to the case of tempered representations. For the transpose-inverse involution of the general linear group, we show that the associated signs are always one.

math.RT↗
arXiv2011

Local analytic conjugacy of resonant analytic mappings in two variables, in the non-archimedean setting

In this note, we consider locally invertible analytic mappings in two dimensions, with coefficients in a non-archimedean field. Suppose such a map has a Jacobian with eigenvalues $λ_1$ and $λ_2$ so that $|λ_1|>1$ and $λ_2$ is a positive power of $λ_1$, or that $λ_1=1$ and $|λ_2|\neq 1$. We prove that two formal maps with eigenvalues satisfying either of these conditions are analytically equivalent if and only if they are formally equivalent.

math.DS↗
arXiv2010

Lipeomorphic equivalence for p-adic analytic functions: a comparison between complex and p-adic dynamics

Let K be a p-adic field, and suppose that f and g are germs of analytic functions on K which are tangent to the identity at 0. It is known that f and g are homeomorphically equivalent, meaning there is an invertible germ h conjugating f to g. In this paper, we determine whether there exists such h which are lipeomorphisms, and moreover find the best possible Holder estimate at 0. Our results have striking complex and real counterparts.

math.DS↗
arXiv2010

Stable Trace Formulas and Discrete Series Representations

Let G be a reductive algebraic group over Q, and suppose that Gamma is an arithmetic subgroup of G(R) defined by congruence conditions. A basic problem in arithmetic is to determine the multiplicities of discrete series representations in L^2(Gamma \ G(R)), and in general to determine the traces of Hecke operators on these spaces. In this paper we give a conjectural formula for the traces of Hecke operators, in terms of stable distributions. It is based on a stable version of Arthur's formula for L^2-Lefschetz numbers, which is due to Kottwitz. We reduce this formula to the computation of elliptic p-adic orbital integrals and the theory of endoscopic transfer. As evidence for this conjecture, we demonstrate the agreement of the central terms of this formula with the unipotent contributions to the multiplicity coming from Selberg's trace formula computed by Wakatsuki, in the case G=GSp_4 and Gamma=GSp_4(Z).

math.NT↗
arXiv2007

Residues of Intertwining Operators for Classical Groups with an Appendix "$L$-Functions and Poles of Intertwining Operators"

Let $\tilde{G}$ be a symplectic or even orthogonal group over a p-adic field $F$, and $M$ the Levi factor of a maximal parabolic subgroup of $\tilde{G}$. Suppose that $M$ has the shape of three blocks of the same size. Let $π$ be a supercuspidal representation of $M$. In this paper we give a simple explicit expression for the residue of the standard intertwining operator for the parabolic induction of $π$ from $M$ to $G$.

math.RT↗
arXiv2007

A p-adic approach to local analytic dynamics: analytic flows and analytic maps tangent to the identity

In this note, we will consider the question of local equivalence of analytic functions which fix the origin and are tangent to the identity, as well as the question of flows of analytic vector fields. All mappings and equivalences are considered in the non-archimedean context e.g. all norms can be considered $p$-adic norms. We show that any two mappings $f$ and $g$ which are formally equivalent are also analytically equivalent, and we show that analytic vector fields generate analytic flows. We consider the related questions of roots and centralizers for analytic mappings. In this setting, anything which can be done formally can also be done analytically.

math.DS↗
arXiv2005

On Arthur's Φ-Function

Write $Θ^E$ for the stable character associated to a finite dimensional representation $E$ of a connected real reductive group $G$. Let $M$ be the centralizer of a maximal torus $T$, and denote by $Φ_M(\gm,Θ^E)$ Arthur's extension of $ |D_M^G(\gm)|^{\half} Θ^E(\gm)$ to $T(\R)$. In this paper we give a simple explicit expression for $Φ_M(\gm,Θ^E)$, when $\gm$ is elliptic in $G$.

math.RT↗