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1. CJM 2014 (vol 67 pp. 184)
Geometric Spectra and Commensurability The work of Reid, Chinburg-Hamilton-Long-Reid,
Prasad-Rapinchuk, and the author with Reid have demonstrated that
geodesics or totally geodesic submanifolds can sometimes be used to
determine the commensurability class of an arithmetic manifold. The
main results of this article show that generalizations of these
results to other arithmetic manifolds will require a wide range of
data. Specifically, we prove that certain incommensurable arithmetic
manifolds arising from the semisimple Lie groups of the form
$(\operatorname{SL}(d,\mathbf{R}))^r \times
(\operatorname{SL}(d,\mathbf{C}))^s$ have the same commensurability
classes of totally geodesic submanifolds coming from a fixed
field. This construction is algebraic and shows the failure of
determining, in general, a central simple algebra from subalgebras
over a fixed field. This, in turn, can be viewed in terms of forms of
$\operatorname{SL}_d$ and the failure of determining the form via certain classes of
algebraic subgroups.
Keywords:arithmetic groups, Brauer groups, arithmetic equivalence, locally symmetric manifolds Category:20G25 |
2. CJM 2010 (vol 62 pp. 1310)
Iwahori--Hecke Algebras of $SL_2$ over $2$-Dimensional Local Fields
In this paper we construct an analogue of Iwahori--Hecke algebras of $\operatorname{SL}_2$ over $2$-dimensional local fields. After considering coset decompositions of double cosets of a Iwahori subgroup, we define a convolution product on the space of certain functions on $\operatorname{SL}_2$, and prove that the product is well-defined, obtaining a Hecke algebra. Then we investigate the structure of the Hecke algebra. We determine the center of the Hecke algebra and consider Iwahori--Matsumoto type relations.
Categories:22E50, 20G25 |
3. CJM 2009 (vol 62 pp. 34)
Branching Rules for Ramified Principal Series Representations of $\mathrm{GL}(3)$ over a $p$-adic Field |
Branching Rules for Ramified Principal Series Representations of $\mathrm{GL}(3)$ over a $p$-adic Field We decompose the restriction of ramified principal series
representations of the $p$-adic group $\mathrm{GL}(3,\mathrm{k})$ to its
maximal compact subgroup $K=\mathrm{GL}(3,R)$. Its decomposition is
dependent on the degree of ramification of the inducing characters and
can be characterized in terms of filtrations of the Iwahori subgroup
in $K$. We establish several irreducibility results and illustrate
the decomposition with some examples.
Keywords:principal series representations, branching rules, maximal compact subgroups, representations of $p$-adic groups Categories:20G25, 20G05 |
4. CJM 2005 (vol 57 pp. 648)
Branching Rules for Principal Series Representations of $SL(2)$ over a $p$-adic Field We explicitly describe the decomposition into irreducibles of
the restriction of the principal
series representations of $SL(2,k)$, for $k$ a $p$-adic field,
to each of its two maximal compact subgroups (up to conjugacy).
We identify these irreducible subrepresentations in the
Kirillov-type classification
of Shalika. We go on to explicitly describe the decomposition
of the reducible principal series of $SL(2,k)$ in terms of the
restrictions of its irreducible constituents to a maximal compact
subgroup.
Keywords:representations of $p$-adic groups, $p$-adic integers, orbit method, $K$-types Categories:20G25, 22E35, 20H25 |