Harmonic analysis studies functions on the real line by expanding them as sums of frequencies (exponentials),

and studying how each term contributes to the overall sum. In many applications, such as speech recognition, only
low frequencies contribute meaningfully.

There is a philosophy, developed by Roger Howe (Yale) over the last 50 years, that this should apply in many other situations.

In a joint work with Howe, we have introduced such a theory for finite classical groups. A
class function on a group can be written as a linear combination of irreducible characters, and we define a notion of “frequency" (aka size) for such objects.  This provides a new way of analyzing the function.

In my talk, I would like to “sell" you on this approach in the first non-trivial example of the group SL(2,F_q) of 2 x 2 matrices with entries in the finite field F_q and determinant equal to one.

The talk should be accessible to anybody who took a good course in linear algebra.

 

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