You are currently browsing the monthly archive for August 2011.

Let and let be the nilpotent radical of the standard Borel subgroup, i.e., is the subgroup of consisting of upper triangular matrices with 1’s on the diagonal. Given a non-trivial character , we define a character of by

This defines a one-dimensional representation of . One can show that the induced representation is multiplicity-free, i.e., when we decompose into a sum of irreducible constituents,

we have for all . Each constituent is called a *generic* irreducible representation. There are many reasons for calling them *generic*. One is that no matter which non-trivial character we choose, we obtain the same induced representation up to equivalence. One can see this by looking at the induced character values on each conjugacy class. These values will be polynomials in and, once simplified, do not depend on the character . Moreover, this *Gelfand-Graev representation* contains *most* irreducible representations of the group.

To see this, we can compare the number of irreducible constituents of with the number of irreducible representations of . By a standard result in the representation theory of finite groups, the number of irreducible representations of is equal to its number of conjugacy classes.