# Weyl character formula for locally compact Lie groups.

I was just wondering if there exists such a formula. Specifically I need to calculate characters of irreducible representations of GSp$(4,\mathbb{C})$.

I know how to do it for the compact Lie group Sp$(4,\mathbb{C})$. Is there a way to extend this to account for GSp$(4,\mathbb{C})$?

I know the structure of the roots and weights for GSp$(4,\mathbb{C})$ so is it ok to just use the WCF blindly in this situation?

Also what would the Weyl group be?

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What is $GSp_4$? – user38268 Oct 1 '12 at 13:20
Google search will tell you. It is essentially a version of Sp4 with a multiplier on the right hand side (Sp4 corresponds to multiplier 1). – fretty Oct 4 '12 at 6:15
I had to write this because I was on my phone and it wouldn't let me write out the full definition in latex. It is quite a standard Lie group, GSp$(4,\mathbb{C}) = \{A\in \text{GL}(2,\mathbb{C})\,|\,A^{T}JA = \mu(A)J\}$, where $J$ is defined as in Sp$(4,\mathbb{C})$ and $\mu$ is a rational multiplier (well in my case this multiplier will be rational). – fretty Oct 5 '12 at 7:19
Pardon me if I'm misunderstanding something, but can't you just (compactly) induce characters of $\mathfrak{sp}(4,\mathbb{C})$ to $\text{GSp}(4,\mathbb{C})$? – Alexander Gruber Oct 8 '12 at 8:59
I am not sure, I don't know an incredible amount about rep theory. I only know how the story goes roughly for compact Lie groups. Since posting this question I have found in the book 1-2-3 of Modular forms a discription of "compact" and "non-compact" roots. My question is now whether I can just plug these into the WCF and whether the answer is correct. I have done this and basically the denominator changes slightly by introducing the multiplier into some of the factors. – fretty Oct 11 '12 at 6:30