The purpose of this paper is to produce explicit realizations of supercuspidal representations of Sp4(k) where k is a p-adic field with odd residual characteristic. These representations will be constructed using the Weil representation of Sp4(k) associated with a certain 4-dimensional compact orthogonal group OQ over k. The main problem addressed in this paper is the analysis of this representation; we need to find how the supercuspidal summands decompose into irreducible pieces.

The problem of decomposing Weil representations has been studied quite a bit already. The Weil representations of SL2(k) associated to 2-dimensional orthogonal groups were used by Casselman [4] and Shalika [9] to produce all supercuspidals of SL2(k). The explicit formulas for these representations were used by Sally and Shalika ([10]) to compute the characters and finally to write down a Plancherel formula for that group.

Throughout this paper, we work in the PL and pseudosimplicial categories, for which we refer to [17] and [10] respectively. For the graph theory involved see [9].

An h-coloured graph (Γ, γ) is a multigraph Γ = (V(Γ), E(Γ)) regular of degree h, endowed with an edge-coloration γ by h colours. If is the colour set, for each we set

For each set . For n ∊ Z, n ≧ 1, set

Δn will be mostly used to denote the colour set for an (n + 1)-coloured graph.

Several authors have considered eigenvalue problems for differential equations where the eigenvalue parameter also appears in the boundary conditions. Such problems do not appear to arise from any spectral problem associated with a linear operator on a Hilbert space. However, it is possible to reset such problems in this context. This has been done for certain second order cases by Walter [4] using a special measure on the interval in question, and by Fulton [1, 2] using the type of space indicated in the title of this article.

It is our purpose here to consider a general class of operators on L2(I) ⊕ Cr, which are based on a differential expression τ of order n on I. We shall first investigate adjoints, boundary conditions, and self-adjointness for such operators. We shall then show that all eigenvalue problems of the form τy = λy, with boundary conditions which involve λ in a linear fashion, can be reset in the context of such operators.

Let H(U) be the linear space of holomorphic functions defined on the unit disk U endowed with the topology of normal (locally uniform) convergence. For a subset E ⊂ H(U) we denote by Ē the closure of E with respect to the above topology. The topological dual space of H(U) is denoted by H′(U).

Let D, 0 ∊ D, be a simply connected domain in C. The unique univalent conformal mapping ϕ from U onto D, normalized by ϕ(0) = 0 and ϕ′(0) > 0 will be called “the Riemann Mapping onto D”. Let S be the set of all normalized univalent functions

Let M be a von Neumann algebra and let {αt}t∊R be a σ-weakly continuous flow on M; i.e., suppose that {αt}t∊R is a one-parameter group of *-automorphisms of M such that for each ρ in the predual, M∗, of M and for each x ∊ M, the function of t, ρ(αt(x)), is continuous on R. In recent years, considerable attention has been focused on the subspace of M, H∞(α), which is defined to be

where H∞(R) is the classical Hardy space consisting of the boundary values of functions bounded analytic in the upper half-plane. In Theorem 3.15 of [8] it is proved that in fact H∞(α) is a σ-weakly closed subalgebra of M containing the identity operator such that

is σ-weakly dense in M, and such that

In this paper we consider the one dimensional Dirac system

1.1

where αk(x) < 0, λ is a complex spectral parameter, and the remaining coefficients are suitably smooth and real valued. We regard (1.1) as regular at x = a but singular at x = b; in Section 4 we extend our result to problems having two singular endpoints.

Equation (1.1) arises from the three dimensional Dirac equation with spherically symmetric potential, following a separation of variables. For the choices p(x) = k/x, αk(x) = 1,p2(x) = (z/x) + c, p1(x) = (z/x) – c, and appropriate values of the constants, (1.1) is the radial wave equation in relativistic quantum mechanics for a particle in a field of potential V = z/x [17]. Such an equation was studied by Kalf [11] in the context of limit point-limit circle criteria, which is one of the matters we consider here.

Throughout the whole paper G will be a locally compact abelian group with Haar measure m and dual group Ĝ. The difference of two sets A and B will be denoted by A ∼ B, i.e.,

For a function f on G and s ∊ G, the functions f′ and fs will be defined by

In this paper, we extend to non-connected, reductive groups over p-adic field of characteristic zero Harish-Chandra's theorem on the local integrability of characters.

Harish-Chandra's theorem states that the distribution character of an admissible, irreducible representation of a (connected) reductive p-adic group is locally integrable. We show that this extends to any reductive group; just as in the connected case, one even gets a very precise control over the singularities of the character along the singular elements.

As will be seen, the proof in the non-connected case is an easy extension of Harish-Chandra's. The reader may wonder why we have bothered to write its generalization completely. The reason is that the original article [8] does not contain proofs for the crucial lemmas, and this makes it impossible to explain why the theorem extends. Because this result is needed for work of Arthur and the author on base change, it has been thought necessary to give complete arguments.

Results of a representation-theoretic nature have played a major role in topos theory since the beginnings of the subject. For example, Deligne's theorem on coherent toposes, which says that every coherent topos has a continuous embedding into a topos of the form SetI for a discrete set I, is a typical result in the representation theory of toposes. (A continuous functor between toposes is the left adjoint of a geometric morphism. For Grothendieck toposes, it is exactly the same as a continuous functor between them, considered as sites with their canonical topologies. By a continuous functor between sites on left exact categories, we mean a left exact functor taking covers to covers.)

A representation-like result for toposes typically asserts that a topos that satisfies some abstract conditions is related to a topos of some concrete kind; the relation between them is usually an embedding of the first topos in the second (concrete) one, for which the embedding satisfies some additional properties (fullness, etc.).

The theory of induced representations has served as a powerful tool in the computations of algebraic K-theory and L-theory ([2], [7], [4, 5], [9], [10, 11, 12, 13], [14], [17], [18]). In this paper we show how to apply this theory to obtain induction theorems for the higher Whitehead groups of Waldhausen. The same technique applies to the analogs of Whitehead groups in unitary K-theory and in L-theory.

For any ring A with unit, let K(A) be the spectrum of the algebraic K-theory of A ([8, p. 343]). Given a discrete group Γ and a subring R of the rational numbers, Loday defines a map of spectra:

*

where (BΓ) is the classifying space of Γ union with a disjoint base point and RΓ is the group-ring of Γ over R.

Let X be a complete separable metric space, and a family of probability measures on the Borel subsets of X. We say that obeys the large deviation principle (LDP) with a rate function I(·) if there exists a function I(·) from X into [0, ∞] satisfying:

• (i) 0 ≦ I(x) ≦ ∞ for all x ∊ X.

• (ii) I(·) is lower semicontinuous.

• (iii) For each l < ∞ the set {x:I(x) ≦ l} is a compact set in X.

• (iv) For each closed set C ⊂ X

  

• (v) For each open set C ⊂ X