Nadia Larsen (Oslo):
Hecke algebras from induced representations and C*-completions
A group G together with a subgroup H form a Hecke pair if
H ∩ xHx-1 is a finite index subgroup of H for all x in G.
The Hecke algebra, which is a convolution algebra of complex
valued functions on G which are biinvariant with respect to the subgroup
H, was introduced in Bost and Connes's work on phase transitions in
number theory.
A powerful method to study the Hecke algebra consists of constructing a
Schlichting completion. This is a Hecke pair formed from a locally
compact group and a compact open subgroup which densely contains the original
pair, cf. Tzanev. In work of Kaliszewski-Landstad-Quigg the construction
is developed further and is used to study C*-completions of Hecke
algebras.
In the talk I will review these notions and constructions, and I will
report on joint work (in progress) with Magnus Landstad, in which we use
the methods of the Schlichting completion to study Hecke algebras from
induced representations. These are Hecke algebras associated to a Hecke
pair and a representation of the subgroup.