Representation 7r Research Articles

Characters, genericity, and multiplicity one for U(3)

Let r : U ~ C x be a generic character of the unipotent radical U of a Borel subgroup of a quasisplit p-adic group G. The number (0 or 1) of ~-Whittaker models on an admissible irreducible representation 7r of (7 was expressed by Rodier in terms of the limit of values of the trace of 7r at certain measures concentrated near the origin. An analogous statement holds in the twisted case. This twisted analogue is used in (F, p. 47) to provide a local proof of the multiplicity one theorem for U(3). This asserts that each discrete spectrum automorphic representation of the quasisplit unitary group U(3) associated with a quadratic extension E/F of number fields occurs in the discrete spectrum with multiplicity one. It is pointed out in (F, p. 47) that a proof of the twisted analogue of Rodier's theorem does not appear in print. It is then given below. Detailing this proof is necessitated in particular by the fact that the attempt in (F, p. 48) at a global proof of the multiplicity one theorem for U(3), although widely quoted, is incomplete, as we point out here.

Strictly positive definite functions on a compact group

We recognize a result of Schreiner, concerning strictly positive definite functions on a sphere in an Euclidean space, as a generalization of Bochner's theorem for compact groups. The purpose of this note is to provide a rapid proof of a relatively new result of Schreiner, [S], concerning strictly positive definite functions on a sphere in a Euclidean space. It is our pleasure to see that the theory of such functions has found applications in geosciences, and that it is well rooted in the existing mathematical literature. Let G be a compact group, and let H C G be a closed subgroup, such that the quotient G/H is infinite. Let f: G C be a continuous function, invariant under the left and right translations by elements of H. We denote the space of all such functions by C(H\G/H). Following Schreiner, [S], we say that the function f is strictly positive definite if and only if n (1) E cicjf(xix j) > 0 i,j~=l for any finite set {Xl I X2, ..., Xn2} C G such that the cosets x1H, x2H, ..., xnH C G/H are distinct, and any complex numbers c1, c2, ..., cn, not all equal to zero. Let G denote the unitary dual of G, [K, 7.3]. This is the set of equivalence classes of irreducible unitary representations of G. For convenience, we choose an irreducible representation for each such class, and identify G with the set of these representations. Recall the Fourier transform (2) f(ir) =ir(f) = ir(x)f(x) dx (e CZ) where f is any absolutely integrable function on G with respect to the Haar measure dx. Thus each ir(f) is a linear map on the finite dimensional Hilbert space 'H1,, where the representation 7r is realized. Let 'H = {v E R, : ir(h)v = v, h E H} be the subspace of H-fixed vectors. Let (G/H) = {r E C; THH A 0}. By the Robenius reciprocity theorem, (G/H)^ is the subset of G, consisting of representations which occur in L2(G/H) (see [K, 8.4]). Received by the editors December 18, 1998 and, in revised form, August 30, 1999. 1991 Mathematics Subject Classification. Primary 43A35, 43A90, 42A82, 41A05.

A simple construction of Stein’s complementary series representations

We given an elementary construction of Stein's complementary series for GL(2n) over an arbitrary local field F , and determine their restrictions to the mirabolic subgroup P2n _ GL(2n 1, F) K F2nI . Taken together with the results in [S], this allows one to calculate the adduced representation Az for an arbitrary irreducible, unitary representation 7r of GL(n, C) . The purpose of this paper is (1) to give an elementary construction of Stein's complementary series representations [St] for GL(2n , F), and (2) to explicitly identify their restriction to the mirabolic subgroup P2n GL(2n 1, F) x 2n-1 F. The main point is that the Bernstein-Zelevinsky theory of the (highest) derivative [B] gives an answer to (2) for nonarchimedean F (where analogous representations were constructed by Godement); namely that the restriction corresponds to the Stein (Godement) representation for GL(2n 2, F) with the same parameter. For archimedean fields it is not known (at least to the author) how to extend the above arguments to answer (2). However, it is reasonable to expect that the answer should be the same for all fields. Having guessed the answer to (2), it seems clear that one should try an inductive argument (on n ) to prove it. Such an argument is given in ?2 and leads to the construction referred to in (1) above. The interesting feature of this argument is that it uses almost nothing from [B] or [St], and reduces the problem to the representation theory of GL(2, F) where the corresponding facts are well known (and easy to prove). 1. STATEMENT OF THE MAIN THEOREM 1.1. Let F be a local field. We begin by describing the degenerate series studied by Stein and Godement. Received by the editors October 17, 1988 and, in revised form, February 20, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 22E46, 22E45, 22E50.

Convolution operators on groups and multiplier theorems for Hermite and Laguerre expansions

Using harmonic analysis on nilpotent Lie groups the following theorem is proved. Let a sequence { a n } \{ {a_{\text {n}}}\} be defined by a function K ∈ C N ( R + ) K \in {C^N}({{\mathbf {R}}_ + }) such that sup λ > 0 | K ( j ) ( λ ) λ j | > ∞ , j = 0 , 1 , … , N {\sup _{\lambda > 0}}|{K^{(j)}}(\lambda ){\lambda ^j}| > \infty ,j = 0,1, \ldots ,N , for N N sufficiently large, putting a n = K ( | n | + m / 2 ) {a_{\text {n}}} = K(|{\text {n}}| + m/2) . Let φ n {\varphi _{\text {n}}} be either Hermite or Laguerre functions. Then the operator \[ ∑ n ( f , φ n ) φ n → ∑ n a n ( f , φ n ) φ n \sum \limits _{\text {n}} {(f,{\varphi _{\text {n}}}){\varphi _{\text {n}}} \to \sum \limits _{\text {n}} {{a_{\text {n}}}(f,{\varphi _{\text {n}}}){\varphi _{\text {n}}}} } \] is bounded on L p ( R m ) {L^p}\left ( {{\mathbb {R}^m}} \right ) or L p ( R + m ) {L^p}(\mathbb {R}_ + ^m) respectively, 1 > p > ∞ 1 > p > \infty .