Explicit Plancherel Research Articles

Multiplicities and Plancherel formula for the space of nondegenerate Hermitian matrices

This paper contains two results concerning the spectral decomposition, in a broad sense, of the space of nondegenerate Hermitian matrices over a local field of characteristic zero. The first is an explicit Plancherel decomposition of the associated L2 space thus confirming a conjecture of Sakellaridis-Venkatesh in this particular case. The second is a formula for the multiplicities of generic representations in the p-adic case that extends previous work of Feigon-Lapid-Offen. Both results are stated in terms of Arthur-Clozel's quadratic local base-change and the proofs are based on local analogs of two relative trace formulas previously studied by Jacquet and Ye and known as (relative) Kuznetsov trace formulas.

Explicit Plancherel Formula for the Space of Symplectic Forms

Abstract We provide a Plancherel decomposition for the space of symplectic bilinear forms of rank $2n$ over a local non-Archimedean field $F$ in terms of that of $\operatorname{GL}_n(F)$.

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

We are interested in the 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 $k$ and give unified description including dyadic case, which is a continuation of our previous papers on non-dyadic case. The space becomes complicated when $e = v_\pi(2) > 0$. First we introduce a typical spherical function $\omega(x;z)$ on $X$, and study their functional equations, which depend on $m$ and $e$, we give an explicit formula for $\omega(x;z)$, where Hall-Littlewood polynomials of type $C_n$ appear as a main term with different specialization according as $m = 2n$ 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.

SPHERICAL FUNCTIONS ON THE SPACE OF p-ADIC UNITARY HERMITIAN MATRICES

We investigate the space X of unitary hermitian matrices over 𝔭-adic fields through spherical functions. First we consider Cartan decomposition of X, and give precise representatives for fields with odd residual characteristic, i.e. 2 ∉ 𝔭. From Sec. 2.2 till the end of Sec. 4, we assume odd residual characteristic, and give explicit formulas of typical spherical functions on X, where Hall–Littlewood symmetric polynomials of type Cn appear as a main term, parametrization of all the spherical functions. By spherical Fourier transform, we show that the Schwartz space [Formula: see text] is a free Hecke algebra [Formula: see text]-module of rank 2n, where 2n is the size of matrices in X, and give the explicit Plancherel formula on [Formula: see text].

Explicit harmonic and spectral analysis in Bianchi I–VII-type cosmologies

The solvable Bianchi I–VII groups which arise as homogeneity groups in cosmological models are analyzed in a uniform manner. The dual spaces (the equivalence classes of unitary irreducible representations) of these groups are computed explicitly. It is shown how parameterizations of the dual spaces can be chosen to obtain explicit Plancherel formulas. The Laplace operator Δ arising from an arbitrary left invariant Riemannian metric on the group is considered, and its spectrum and eigenfunctions are given explicitly in terms of that metric. The spectral Fourier transform is given by means of the eigenfunction expansion of Δ. The adjoint action of the group automorphisms on the dual spaces is considered. It is shown that Bianchi I–VII-type cosmological spacetimes are well suited for mode decomposition. The example of the mode decomposed Klein–Gordon field on these spacetimes is demonstrated as an application.

Stepwise square integrable representations of nilpotent Lie groups

We study the conditions for a nilpotent Lie group to be foliated into subgroups that have square integrable (relative discrete series) unitary representations, that fit together to form a filtration by normal subgroups. Then we use that filtration to construct a class of “stepwise square integrable” representations on which Plancherel measure is concentrated. Further, we work out the character formulae for those stepwise square integrable representations, and we give an explicit Plancherel formula. Next, we use some structure theory to check that all these constructions and results apply to nilradicals of minimal parabolic subgroups of real reductive Lie groups. Finally, we develop multiplicity formulae for compact quotients $$N/\varGamma $$ where $$\varGamma $$ respects the filtration.

Plancherel measure for [formula omitted] and [formula omitted]: Explicit formulas and Bernstein decomposition

Let F be a nonarchimedean local field, let D be a division algebra over F, let GL ( n ) = GL ( n , F ) . Let ν denote Plancherel measure for GL ( n ) . Let Ω be a component in the Bernstein variety Ω ( GL ( n ) ) . Then Ω yields its fundamental invariants: the cardinality q of the residue field of F, the sizes m 1 , … , m t , exponents e 1 , … , e t , torsion numbers r 1 , … , r t , formal degrees d 1 , … , d t and conductors f 11 , … , f tt . We provide explicit formulas for the Bernstein component ν Ω of Plancherel measure in terms of the fundamental invariants. We prove a transfer-of-measure formula for GL ( n ) and establish some new formal degree formulas. We derive, via the Jacquet–Langlands correspondence, the explicit Plancherel formula for GL ( m , D ) .

Approximating Fourier transformation of orbital integrals

In this paper, we are concerned with orbital integrals on a class C of real reductive Lie groups with non-compact Iwasawa K-component. The class C contains all connected semisimple Lie groups with infinite center. We establish that any given orbital integral over general orbits with compactly supported continuous functions for a group G in C is convergent. Moreover, it is essentially the limit of corresponding orbital integrals for its quotient groups in Harish-Chandra's class. Thus the study of orbital integrals for groups in class C reduces to those of Harish-Chandra's class. The abstract theory for this limiting technique is developed in the general context of locally compact groups and linear functionals arising from orbital integrals. We point out that the abstract theory can be modified easily to include weighted orbital integrals as well. As an application of this limiting technique, we deduce the explicit Plancherel formula for any group in class C .

Relative spherical functions onp-adic symmetric spaces (three cases)

Let F be a non-archimedean local field with residual field of odd characteristic. Given a reductive group G defined over F, equipped with an involution denoted g → g*, let K be a maximal compact of G. G acts on the space {x E G |x = x*} by g . x = g x g*. Let s 0 ∈ G be fixed by the involution and let S = G . s 0 and H = Stab G (s 0 ). A relative spherical function on S is a K-invariant function on S, which is an eigenfunction of the Hecke algebra of G relative to K. The problem at hand is to classify all such functions, compute them explicitly in terms of Macdonald polynomials and obtain an explicit Plancherel measure. We obtain a complete solution in three cases relevant to the theory of Automorphic Forms. Namely: Case 1: G = GL (2n, F), H = GL (n, F) × GL (n, F). Case 2: G = GL (m, E), H = GL (m, F). Case 3: G = GL (2n, F), H = GL (n, E). E is an unramified quadratic extension of F.

Explicit Plancherel formula for the p-adic group GL( n)

We provide an explicit Plancherel formula for the p-adic group GL( n). We determine explicitly the Bernstein decomposition of Plancherel measure, including all numerical constants. We also prove a transfer-of-measure formula for GL( n). To cite this article: A.-M. Aubert, R. Plymen, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Beta functions of Bruhat-Tits buildings and deformation of on the set of -adic lattices

For the space of all lattices in an -dimensional -adic linear space an analogue of the matrix beta function is constructed; this beta function can degenerate to the Tamagawa zeta function. An analogue of Berezin kernels for is proposed. Conditions for the positive-definiteness of these kernels and an explicit Plancherel's formula are obtained.

Beta functions of Bruhat-Tits buildings and deformation of l{sup 2} on the set of p-adic lattices

For the space Lat{sub n} of all lattices in an n-dimensional p-adic linear space an analogue of the matrix beta function is constructed; this beta function can degenerate to the Tamagawa zeta function. An analogue of Berezin kernels for Lat{sub n} is proposed. Conditions for the positive-definiteness of these kernels and an explicit Plancherel's formula are obtained.

Plancherel Formula for Berezin Deformation of L2 on Riemannian Symmetric Space

Consider natural representations of the pseudounitary group U(p, q) in the space of holomorphic functions on the Cartan domain (Hermitian symmetric space) U(p, q)/(U(p)×U(q)). Berezin representations of O(p, q) are the restrictions of such representations to the subgroup O(p, q). We obtain the explicit Plancherel formula for the Berezin representations. The support of the Plancherel measure is a union of many series of representations. The density of the Plancherel measure on each piece of the support is an explicit product of Γ-functions. We also show that the Berezin representations give an interpolation between L2 on noncompact symmetric space O(p, q)/O(p)×O(q) and L2 on compact symmetric space O(p+q)/O(p)×O(q).

An explicit Plancherel formula for completely solvable Lie groups.

An explicit Plancherel formula for completely solvable Lie groups.

The plancherel formula for the infinite XXZ Heisenberg spin chain

In this Letter we discuss the explicit Plancherel formula for the Bethe Ansatz eigenstates for the Hamiltonian of the infinite XXZ Heisenberg-Ising spin chain in the N-magnon sectors: N=2, 3,... In particular, we shall point out that a striking spectral phenomenon occurs when the coupling constant c is such that 0<|c|<1.

A Plancherel formula for parabolic subgroups

We obtain explicit Plancherel formulas for the parabolic subgroups P P of p p -adic unitary groups which fix one dimensional isotropic subspaces. By means of certain limits of difference operators (called strong derivatives), we construct a Dixmier-Pukanszky operator which compensates for the nonunimodularity of the group P P . Then, we compute the Plancherel formula of N ⋅ A N \cdot A , where N N is the nilradical of P P and A = Q p ′ A = {Q’_p} , the multiplicative group of nonzero p p -adic numbers, by formulating a p p -adic change of variable formula and using the strong derivative.

Spherical functions on Cartan motion groups

This paper gives a reasonably complete treatment of harmonic analysis on Cartan motion groups. Included is an explicit parameterization of irreducible spherical functions of general K K -type, and of the nonunitary dual (and its topology). Also included is the explicit Plancherel measure, the Paley Wiener theorem, and an asymptotic expansion of general matrix entries. (These are generalized Bessel functions.) However the main result is Theorem 19, a technical result which measures the size of the centralizer of K K in the universal enveloping algebra of the corresponding reductive group.

Canonical semi-invariants and the Plancherel formula for parabolic groups

A parabolic subgroup of a reductive Lie group is called "good" if the center of the universal enveloping algebra of its nilradical contains an element that is semi-invariant of weight proportional to the modular function. The "good" case is characterized here by invariance of the set of simple roots defining the parabolic, under the negative of the opposition element of the Weyl group. In the "good" case, the unbounded Dixmier-Pukanszky operator of the parabolic subgroup is described, the conditions under which it is a differential operator rather than just a pseudodifferential operator are specified, and an explicit Plancherel formula is derived for that parabolic.

Canonical Semi-Invariants and the Plancherel Formula for Parabolic Groups

A parabolic subgroup of a reductive Lie group is called "good" if the center of the universal enveloping algebra of its nilradical contains an element that is semi-invariant of weight proportional to the modular function. The "good" case is characterized here by invariance of the set of simple roots defining the parabolic, under the negative of the opposition element of the Weyl group. In the "good" case, the unbounded Dixmier-Pukanszky operator of the parabolic subgroup is described, the conditions under which it is a differential operator rather than just a pseudodifferential operator are specified, and an explicit Plancherel formula is derived for that parabolic.

The Plancherel Formula for parabolic subgroups

We prove an explicit Plancherel Formula for the parabolic subgroups of the simple Lie groups of real rank one. The key point of the formula is that the operator which compensates lack of unimodularity is given, not as a family of implicitly defined operators on the representation spaces, but rather as an explicit pseudo-differential operator on the group itself. That operator is a fractional power of the Laplacian of the center of the unipotent radical, and the proof of our formula is based on the study of its analytic properties and its interaction with the group operations.