## Tokyo Journal of Mathematics

- Tokyo J. Math.
- Volume 40, Number 2 (2017), 517-564.

### Harmonic Analysis on the Space of $p$-adic Unitary Hermitian Matrices, Mainly for Dyadic Case

#### Abstract

We are interested in harmonic analysis on $p$-adic homogeneous spaces based on spherical functions. In the present paper, we investigate the space $X$ of unitary hermitian matrices of size $m$ over a ${\mathfrak p}$-adic field $k$ mainly for dyadic case, and give the unified description with our previous papers for non-dyadic case. The space becomes complicated for dyadic case, and the set of integral elements in $X$ has plural Cartan orbits. We introduce a typical spherical function $\omega(x;z)$ on $X$, study its functional equations, which depend on $m$ and the ramification index $e$ of $2$ in $k$, and give its explicit formula, where Hall-Littlewood polynomials of type $C_n$ appear as a main term with different specialization according as the parity $m = 2n$ or $2n+1$, but independent of $e$. By spherical transform, we show the Schwartz space ${\mathcal S}(K\backslash X)$ is a free Hecke algebra ${\mathcal H}(G,K)$-module of rank $2^n$, and give parametrization of all the spherical functions on $X$ and the explicit Plancherel formula on ${\mathcal S}(K\backslash X)$. The Plancherel measure does not depend on $e$, but the normalization of $G$-invariant measure on $X$ depends.

#### Article information

**Source**

Tokyo J. Math., Volume 40, Number 2 (2017), 517-564.

**Dates**

First available in Project Euclid: 9 January 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.tjm/1515466838

**Digital Object Identifier**

doi:10.3836/tjm/1502179240

**Mathematical Reviews number (MathSciNet)**

MR3743731

**Zentralblatt MATH identifier**

06855947

**Subjects**

Primary: 11E85

Secondary: 11E95: $p$-adic theory 11F70: Representation-theoretic methods; automorphic representations over local and global fields 22E50: Representations of Lie and linear algebraic groups over local fields [See also 20G05] 33D52: Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.)

#### Citation

HIRONAKA, Yumiko. Harmonic Analysis on the Space of $p$-adic Unitary Hermitian Matrices, Mainly for Dyadic Case. Tokyo J. Math. 40 (2017), no. 2, 517--564. doi:10.3836/tjm/1502179240. https://projecteuclid.org/euclid.tjm/1515466838