I played with it for GL_3. The difference above is twice the Steinberg. If I compute for GL_3 the "analogous" sum A + 2 C - 3 B, where A, B, C are induced from the rank 3, 2, 1 tori respectively, then I get a representation of dimension 6q^3, and I think its character values agree with 6*Steinberg on regular semisimple elements. Of course it might not actually be..

Historical comment (since you say "a particular case studied by Hecke...") : I doubt that Hecke studied the particular algebra you mention; he was studying the convolution algebra corresponding to classical "Hecke operators" in the theory of modular forms. Incidentally, these were in fact introduced by Mordell.

This is a very satisfactory answer, thanks.

Do you mean an n x n matrix? If so, the probability of being singular seems to be 1 as formulated in the first paragraph: Each row has a positive probability of being identically zero. Perhaps the diagonal changes this?

Oh, I see what you mean. Thanks for clarifying!

\mu_2 is not etale over Spec(Z).