(July 2, 2005) Irreducibles as kernels of intertwinings among principal series Paul Garrett
[email protected] http://www.math.umn.edu/˜garrett/ Let G = SL(2, R). Our goal is to find irreducible representations of G. The actual work we do is quite prosaic, and we do find several sorts of representations of G, without any fancier ideas. However, the completeness of our computation depends on some more serious results. Further, the motivation for this computation, and the idea that it would be fruitful, come from less pedestrian thinking. The computation itself can be understood on its own terms without understanding the broader context. For that matter, the fact that a few simple if non-elementary ideas lead to such tangible computational results might be construed as a motivation to study those non-elementary notions. To understand the larger context, first note that Casselman’s subrepresentation theorem asserts that any irreducible representation[ ] π of G is a subrepresentation of some one of the principal series representations Is defined below. [ ] Further, the quotient Is /π (or a further quotient that is irreducible) again imbeds in some Is0 with another parameter value s0 . Thus, any irreducible π appears as a (possibly subrepresentation of a) kernel of a G-homomorphism Is −→ Is0 among principal series. Still further, examination of the eigenvalues of the center of the enveloping algebra [ ] on the Is shows that the only values s0 such that Is0 has a non-trivial G-homorphism Is −→ Is0 are s0 = s and s0 = 1 − s. There is a natural integral for a G-homomorphism Is −→ Is0 , which can be evaluated in terms of the gamma function on adroitly chosen vectors in the principal series. But, again, the computation itself is understandable without necessarily fully appreciating this grounding of it. This computation also gives an approach to understand why the gamma function should not vanish. • Principal series representations • The main computation • Subrepresentations • Return to smooth vectors • Appendix: usual tricks with Γ(s) [ ] To be more accurate, the subrepresentation theorem in fact asserts the imbeddability of irreducible (g, K) representations, also called (g, K)-modules. The aptness of this notion was one of Harish-Chandra’s basic and indispensable contributions to this subject. Here g is the Lie algebra of G acting by the differentiated version of the action of G (at least on smooth vectors of the representation), and K is a maximal compact subgroup of G. Thus, the (g, K)-module structure forgets some of the structure of a G-representation. [ ] Actually, our present discussion only discusses half the principal series, namely the even or unramified ones. [ ] The enveloping algebra is the associative algebra generated by g with relations xy − yx = [x, y] for x, y ∈ g. The structure of its center is described by an early theorem of Harish-Chandra, and by a Schur lemma the center acts by scalars on irreducibles. Further, it is our good fortune that the eigenvalues of the center quite successfully distinguish among principal series. 1 Paul Garrett: Kernels of intertwining operators (July 2, 2005) 1. Principal series representations Define useful subgroups [ ] of G by 1 x N = {nx = : x ∈ R} 0 1 and [ a 0 M ={ 0 1/a : a ∈ R× } ] P = NM = MN The unramified principal series representation[ prescribed left equivariance ] Is is as a space of smooth[ ] functions f on G with the Is = {f : f (nmg) = χs (p) f (g) for all p ∈ P, g ∈ G} where s ∈ C and χs