It is shown that any $\lambda$-firmly, $0 < \lambda < 1$ , nonexpansive mapping $T:C \to C$ has a fixed point in $C$ whenever $C$ is a finite union of nonempty, bounded, closed convex subsets of a uniformly convex Banach space.
This is an announcement of a new result which is a generalization of Popa’s result in [Po]. Popa gives explicit formulas of Whittaker functions on GL(2,C) at the minimal K-types, and shows that the local zeta integrals for GL(2,C) defined from some Whittaker functions are equal to the associated L-factors. In this article, we will give explicit formulas of Whittaker functions on GL(2,C) at all K-types, and show that the local zeta integrals for GL(2,C) × GL(2,C) defined from some Whittaker functions are equal to the associated L-factors. Proofs will appear elsewhere. 1. Notation Let G = GL(2,C) be the complex general linear group of degree 2. In this article, we regard G as a real reductive Lie group. We fix an Iwasawa decomposition G = NAK of G with N = {( 1 x 0 1 x ∈ C } , A = {( y1y2 0 0 y2 y1, y2 ∈ R>0 } , K = U(2). Let g be the associated Lie algebra of G. We denote by gC the complexification g⊗R C of g. It is known that gC is isomorphic to gl(2,C)⊕ gl(2,C) via gC ∋ X ⊗ t → (tX, tX) ∈ gl(2,C)⊕ gl(2,C). Here X means the complex conjugate of X. We denote by U(gC) the universal enveloping algebra of gC, and by Z(gC) the center of U(gC). We note that U(gC) ≃ U(gl(2,C)⊕ gl(2,C)) ≃ U(gl(2,C))⊗C U(gl(2,C)). Here the second isomorphism above is induced from gl(2,C)⊕ gl(2,C) ∋ (X1, X2) → X1 ⊗ 1 + 1⊗X2 ∈ U(gl(2,C))⊗C U(gl(2,C)). Let Λ = {(λ1, λ2) ∈ Z | λ1 ≥ λ2}. For λ = (λ1, λ2) ∈ Λ, let Vλ be the C-vector space of degree λ1−λ2 homogeneous polynomials in two variables z1, z2, on which 2010 Mathematics Subject Classification. Primary 11F66, Secondary 11F30
In this article the author presents the results on the explicit construction of the intertwining operator between a holomorphic discrete series representation of some Lie group G and that of some subgroup G1 ⊂ G. More precisely, we construct a G1-intertwining projection operator from a representation H of G onto a representation H1 of G1 as a differential operator, in the case (G,G1) = (G0 ×G0,∆G0) and both H, H1 are of scalar type, and also construct a G1-intertwining embedding operator from H1 of G1 into H of G as an infinite-order differential operator, in the case H is of scalar type and H1 is multiplicity-free under a maximal compact subgroup K1 ⊂ G1. In this paper we mainly deal with the case (G,G1) = (Sp(1,R)×Sp(1,R),∆Sp(1,R)) and the cases (G,G1) = (SU(s, s), Sp(s,R)), (SU(s, s), SO∗(2s)).
In this paper we propose similarity between ramified irregular singularities of meromorphic connections on formal disk and plane curve singularities. First we relate Komatsu-Malgrange irregularities of meromorphic connections to intersection numbers and Milnor numbers of plane curve germs. Next we see that local Fourier transforms of connections can be seen as blow up of plane curves. Moreover a necessary and sufficient condition for an irreducible connection to have a resolution of the ramified singularity is determined as an analogy of the resolution of plane curve singularities. Finally, for meromorphic connections we define an analogue of Puiseux characteristics which are topological invariants of plane curve singularities and show that it can be seen as an invariant of Stokes structures of meromorphic connections. Introduction Our interest in this paper is the similarity between ramified irregular singularities of meromorphic connections on formal disk and plane curve singularities. Meromorphic connections can be seen as modules over a “non-commutative” ring, the ring of differential operators, and plane curve germs have “commutative” rings as their local rings, stalks of structure sheaves. Between these commutative and non-commutative ones, we shall find similarities of (i) invariants: intersection numbers and Milnor numbers of curves and Komatsu-Malgrange irregularities of connections, of (ii) transformations: the blow up of curves and the local Fourier transform of connections, and of (iii) topological structures: knots arising from curve singularities and Stokes structures of connections. To state our main theorems, we recall some definitions which are explained in detail in the latter sections. Let K be an algebraically closed field of characteristic zero. For a positive integer q and f ∈ K((x 1 q )) with −p/q = ord(f) < 0, let us define Ef,q = (V,∇), a connection over K((x)), as follows. Regard V = K((x 1 q )) as The author is supported by JSPS Grant-in-Aid for Young Scientists (B) Grant Number 26800072
