Chal Benson

On Gelfand pairs associated with solvable Lie groups

Let G be a locally compact group, and let K be a compact subgroup of Aut(G) , the group of automorphisms of G. There is a natural action of K on the convolution algebra L (G), and we denote by LK(G) the subalgebra of those elements in L (G) that are invariant under this action. The pair (K, G) is called a Gelfand pair if LI(G) is commutative. In this paper we consider the case where G is a connected, simply connected solvable Lie group and K C Aut(G) is a compact, connected group. We characterize such Gelfand pairs (K, G), and determine a moduli space for the associated K-spherical functions.

Rationality of the generalized binomial coefficients for a multiplicity free action

Let V be a finite dimensional Hermitian vector space and K be a compact Lie subgroup of U(V ) for which the representation of K on C[V ] is multiplicity free. One obtains a canonical basis {pα} for the space C[VR] ofK-invariant polynomials on VR and also a basis {qα} via orthogonalization of the pα’s. The polynomial pα yields the homogeneous component of highest degree in qα. The coefficients that express the qα’s in terms of the pβ ’s are the generalized binomial coefficients of Yan. The main result in this paper shows that these numbers are rational.

On Multiplicity Free Actions

1. Preliminaries 1 2. Multiplicity free actions 8 3. Linear multiplicity free actions 15 4. Examples of multiplicity free decompositions 21 5. A recursive criterion for multiplicity free actions 33 6. The classification of linear multiplicity free actions 37 7. Invariant polynomials and differential operators 43 8. Generalized binomial coefficients 52 9. Eigenvalues for operators in PD(V ) 61 References 67

Three-step Harmonic Solvmanifolds

The Lichnerowicz conjecture asserted that every harmonic Riemannian manifold is locally isometric to a two-point homogeneous space. In 1992, E. Damek and F. Ricci produced a family of counter-examples to this conjecture, which arise as abelian extensions of two-step nilpotent groups of type-H. In this paper we consider a broader class of Riemannian manifolds: solvmanifolds of Iwasawa type with algebraic rank one and two-step nilradical. Our main result shows that the Damek–Ricci spaces are the only harmonic manifolds of this type.

The moment map for a multiplicity free action

Let K be a compact connected Lie group acting unitarily on a finite-dimensional complex vector space V. One calls this a multiplicity-free action whenever the AT-isotypic components of C[V] are A-irreducible. We have shown that this is the case if and only if the moment map t : V —► t* for the action is finite-to-one on A-orbits. This is equivalent to a result concerning Gelfand pairs associated with Heisenberg groups that is motivated by the Orbit Method. Further details of this work will be published elsewhere.

Gelfand Pairs Associated with Finite Heisenberg Groups

We examine a family of finite Gelfand pairs which arise in connection with Heisenberg groups H = H n (F) over finite fields of odd characteristic. The symplectic group Sp(n, F) acts on H by automorphisms. A subgroup K of Sp(n, F) yields a Gelfand pair (K, H) when the K-invariant functions on H commute under convolution. This is equivalent to the restriction of the oscillator representation to K being multiplicity free. An interesting example of this type occurs with K a finite analog of the unitary group U(n).