Aarhus University Seal / Aarhus Universitets segl

Calculation of the distributional-character $\pi_{\mathrm{met}}$ of $\mathrm{SL}(2, \mathbb{R})$

Adam Ehlers Nyholm Thomsen
Foredrag for studerende
Fredag, 3 oktober, 2014, at 15:15-16:00, in Aud. D4 (1531-219)
Abstrakt:
The metaplectic representaion (also called Segal-Shale-Weil or
oscillator) is a family of representations of double covers of the
symplectic groups.  One way of realising the metaplectic representation
is through the Stone-von-Neumann theorem classifying the representations
of the Heisenberg group.  The metaplectic representation is a direct sum
of two irreducible subrepresentations the even and the odd part both of
which are so-called minimal representations; that is, their associated
ideal is the Joseph ideal or equivalently their Gelfand-Kirillov
dimension is minimal.  The distributional character of a representation
is essentially a method for taking a trace of an infinite dimensional
representation, and by a theorem of Harish-Chandra it is a locally
integrable function on the group.  In this talk I will show how we can
reduce the calculation of the distributional character of the
metaplectic representation of the double cover $\tilde{\mathrm{SL}}(2,\mathbb{R})$ of $\mathrm{SL}(2, \mathbb{R})$ by reducing it to a
geometric sum and a result on the eigenvalues of the elements of the
compression semigroup of the upper-halfplane.

In this talk I intend to give an introduction to a small section of the
representation theory of semisimple Lie groups, through the example of
the metaplectic representation of $\tilde{\mathrm{SL}}(2, \mathbb{R})$.
I intend for the talk to be quite elementary in its approach; hence I
will only assume familiarity with complex analysis, group theory (the
definition of a group and group actions) and linear algebra.
Kontaktperson: Thomas Lundsgaard Schmidt