Moscow-Petersburg seminar on low-dimensional mathematics
January 9, 2004. 4:00pm. PDMI, Fontanka 27, room 311.

D. A. Timashev (Moscow State University)
Complexity of homogeneous spaces and growth of multiplicities of
irreducible representations

The complexity of a homogeneous space G/H under a connected
reductive (e.g. semisimple) group G is by definition the
codimension of general orbits of a Borel subgroup B or, in other
words, the maximal number of algebraically independent
B-invariant rational functions on G/H. This geometric invariant
plays an important role in the geometry and harmonic analysis
on homogeneous spaces. For instance, it controls the theory of
equivariant embeddings of G/H. In this talk, we give a
representation-theoretic interpretation of this number as the
exponent of growth for multiplicities of simple G-modules in the
algebra of regular functions on G/H or, more generally, in the
spaces of sections of homogeneous line bundles. The idea of the
proof is to show that these multiplicities are bounded from
above by the dimensions of certain Demazure modules. This
estimate for multiplicities is uniform, i.e., it does not
depend on G/H.

For homogeneous spaces of complexity 0 or 1 a precise formula
for multiplicities can be obtained. (This justifies the term
"complexity".) In particular, for homogeneous spaces of
complexity 0, called spherical (including symmetric spaces and
projective homogeneous spaces), all multiplicities are equal to
0 or 1 (Vinberg, Kimelfeld).

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http://www.pdmi.ras.ru/~lowdimma