# Regular semisimple elements in $SL(n,q)$

Consider $$G=GL(n,q)$$. A regular semisimple element of this group, is a matrix, whose characterestic polynomial is square-free and same as minimal polynomial. Results show that the number of such matrices in $$G$$ is a polynomial in $$q$$, with degree $$n^2$$, and coefficient of the leading term being 1.

Does similar hold, when we take $$G=SL(n,q)$$, that is, the number of such matrices in $$G$$ is given by a polynomial in $$q$$ of degree $$n^2-1$$, with the leading term being 1?

I appreciate any kind of help or references that can be of use. Thank you.

• If n=2 then the number of non-regular-semisimple elements is q^2-q+1 for odd q and q^2 for even q. So what sort of answer are you looking for? Do you care if there is a different formula for a few small characteristics? I expect the general formula is of similar qualitative behaviour to this case. – Peter McNamara Apr 16 at 1:01
• @PeterMcNamara, I'm not familiar with these counting problems. Does that mean that the statement in the question about the number of rss elements in $\operatorname{GL}(n, q)$ is also true only for large $q$, or does passing to $\operatorname{GL}(n, q)$ magically sort out $\operatorname{SL}(n, q)$ problems? – LSpice Apr 16 at 1:07
• I can't find a reference on the number of rss elements, but the number of rss classes is well studied. I'm familiar with Steinberg (see, for example, §14 of Steinberg - Endomorphisms of linear algebraic groups); but some Googling also turned up Fleischmann, Janisczczak, and Knörr - The number of regular semisimple classes of special linear and unitary groups. – LSpice Apr 16 at 1:27
• @LSpice I suspect that for SL_N one will get a different formula for q a power of a prime dividing N, and this phenomenon does not appear for GL_N, where the OP indicates there is a general formula that works for all q. – Peter McNamara Apr 16 at 1:53
• number of regular semisimple elements is always asymptotic to q^{n^2-1} by Deligne's proof of Weil conjectures. – Peter McNamara Apr 17 at 0:58