Vijest
NEW DOCTORAL DEGREE Unitary duals of p-adic groups SO(5) and Mp(2)
Ivan Matić
orcid.org/0000-0001-9264-9293
; Departemnt of Mathematics, University of Osijek, Osijek, Croatia
Sažetak
The problem of the determination of the unitary dual of a reductive algebraic group
is one of the most important problems in representation theory,
with numerous applications in harmonic analysis and the theory of
automorphic forms. In this thesis, we classify the unitary duals
of some low-rank reductive $p$-adic groups.
In the initial chapter, we systematically present the
classification of the unitary dual of the group $SO(5, F)$
(modulo cuspidal representations), where $F$ is a local
non-archimedean field. We
use the method of Jacquet modules and intertwining operator
methods.
In the next chapter, we determine non-cuspidal part of the
unitary dual of the non-linear metaplectic group $Mp(2)$, an
unique nontrivial two-fold cover of the group $Sp(2)$ over a
local non-archimedean field of characteristic different than two.
We rely on the new results of authors Marcela Hanzer and Goran
Mui\'{c}, where they have related reducibilities of the
representations of metaplectic groups with those of the special
odd orthogonal groups, using theta correspondence. Besides that,
those results allow us to use the method of Jacquet modules in
investigating the composition series of the representations of
the metaplectic group. In cases when this powerful methods happen
to be insufficient tool, we use the theory of theta
correspondence for dual pairs $Mp(n) \times O(2r+1)$.
Generalizing some of the results obtained while determining the
unitary dual of the group $Mp(2)$, in the following chapter we
prove irreducibility of the unitary principal series for $Mp(n)$.
Let $Spin(2n+1)$ denote the simply-connected algebraic group of
the type $B_{n}$, which can also be obtained as a two-fold cover
of the classical group $SO(2n+1)$ if we are working over an
algebraically closed field. In the representation theory, it is
very important to know what the Levi subgroups in considered group
look like, especially when investigating the parabolically induced
representations. As a contribution to the study of this problem, in the final chapter we give
a precise description of Levi subgroups of the group $Spin(2n+1)$
over algebraic closure of a $p$-adic field.
Ključne riječi
Hrčak ID:
53681
URI
Datum izdavanja:
10.6.2010.
Posjeta: 1.157 *