11:32a.m. March 15, 2012

D-modules and representation theory

Bill Casselman University of British Columbia [email protected]

This records my attempt to understand the role of D•modules in representation theory. My main reference has been the short survey [Miliˇci´c:1993]. Let G be a connected complex reductive group, K the subgroup fixed by an involution. These are derived from a reductive group GR defined over R and a maximal compact subgroup KR. Let X be the flag variety of G, parametrizing its Borel subgroups. The basic idea is to explain the dictionary between what Miliˇci´ccalls Harish-Chandra sheaves on X and Harish•Chandra modules of (G,K). Without details, I recall that a Harish•Chandra sheaf is a coherent sheaf F with support on a K•orbit Y in X, together with a K•equivariant connection τ on that orbit. These data determine the structure of a Dλ•module on some space associated to the pair (Y, τ), with support on the closure of Y . Such connections (and the vector bundle of the connection) sheaves are determined by a finite•dimensional representation of π0(Ky), where Ky is the isotropy subgroup of K fixing y ∈ Y . The ring sheaf Dλ is the twisted ring corresponding to the charcatrer of H. I’ll explain these notions in detail later on.

1. Tori

I’ll look here at three of the simplest examples. × × Example. Let GR be the group of z in C with |z| = 1. Then G itself is just C , and K = G. The flag manifold is a single point. I think H must be G itself, similarly for λ because we require compatibility of the actions. The isotropy group is K itself, so the connections we want are just the characters of K. × × λ m Example. Let GR = R , K = {±1}. Then G = C . We should get all the characters |x| sgn (x) of GR, but how? I think I have to take X again to be a point. The connections correspond to the two characters sgnm of K. The group H is G itself? The characters sgnm of K give us the sign factor, and λ comes from the twisted D•ring. So it must be that I take X = G/K? With twisted operators D − λ where D = zd/dz? × λ n Example. Let GR = C . Characters |z| z . These three examples essentially exhaust the possibilities, since every real torus is a product of these. So . . . if G is a torus X is always a single point? The K•connections are just the sign’ representations of the split factors, and H = G itself, with K•compatibility forcing λ to be of the right form?

2. The group SL2(R) D•modules and representation theory (11:32 a.m. March 15, 2012) 2

3. References

1. Dragan Miliˇci´c, ‘Algebraic D•modules and representation theory of semi•simple Lie groups’, Con• temporary Mathematics 154 (1993), 133–168. 2. Ryoshi Hotta, Kiyoshi Takeuchi, and Toshiyuki Tanisaki, D-modules, perverse sheaves, and repre- sentation theory, Progress in Mathematics 236, Birtkhauser,¨ 2000. 3. Alexander Beilinson and Joseph Bernstein, ‘localisation de g•modules’, Comptes Rendues de l’Academie´ Scientifique de Paris 292 (1981), 15–18. 4. Armand Borel et al., Algebraic D-modules, Academic Press, 1987. 5. Joseph Bernstein, ‘Algebraic theory of D•modules’, lecture notes, 1985.