Guide to proof of minimal K type formula

Let G be a linear connected reductive group.  If K is a maximal compact subgroup of G, the published paper "Minimal K-type formula" gives a formula for the minimal K types of all standard induced representations of G.  A proof of the published formula appears in the two sets of accompanying handwritten notes.  The paper and the notes say "semisimple" instead of "reductive," but it is tidier to work in the more general situation "reductive," and the notes in effect do that.

The two sets of accompanying notes were written probably between 1980 and 1982.  One set, reproduced as "EqualRankCase.pdf," is 82 pages long and handles the case that rank G = rank K.  The other set, reproduced as "ReductionToEqualRank.pdf," is 10 pages long and reduces the general case to the equal rank case.  The specific reduction is from a general G to the centralizer in G of a maximal torus in K.  This centralizer is a linear connected reductive group of equal-rank type.

One detail of structure theory, namely the connectedness of the above centralizer, was suspected by the author at the time of the writing, and a proof is included in the author's book "Lie Groups Beyond an Introduction," published in a first edition in 1996 and in a second edition in 2002.

The notes in the equal-rank case make use of the notion of a "fine" K type for a split linear group.  This notion was studied by D. A. Vogan in an unpublished preprint from the 1970s entitled "Fine K types and the principal series."  Page 35 of the first set of notes summarizes the definition and  properties of these representations that are used in the notes.