Discrete Series Representation - Properties

Properties

If G is unimodular, an irreducible unitary representation ρ of G is in the discrete series if and only if one (and hence all) matrix coefficient

with v, w non-zero vectors is square-integrable on G, with respect to Haar measure.

When G is unimodular, the discrete series representation has a formal dimension d, with the property that

\displaystyle d\int \langle \rho(g)\cdot v, w \rangle \overline{\langle \rho(g)\cdot x, y \rangle}dg =\langle v, x \rangle\overline{\langle w, y \rangle}

for v, w, x, y in the representation. When G is compact this coincides with the dimension when the Haar measure on G is normalized so that G has measure 1.

Read more about this topic:  Discrete Series Representation

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