Guide to proof of minimal K type
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
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
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Vogan in an unpublished preprint from the 1970s entitled "Fine K types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.