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Speaker: Alexandre Zalesski (Minsk)

Title: Singer Tori in Representations of \(PGL_n(q)\)

Abstract:

Let \(G=PGL(n,q)\), and let \(T\) be a subgroup of \(G\) of order \((q^n-1)/(q-1)\), whose preimage in \(GL(n,q)\) is irreducible. Let \(\chi\) be an irreducible character of \(G\) of degree greater than \(1\). Then one of the following holds:

(1) \(1_T\) is a constituent of the restriction \(\chi|_T\);

(2) \(\chi(1)=(q^n-q)/(q-1)\);

(3) \((n,q)=(3,2)\) and \(\chi(1)=3\).

This answers a question of Pablo Spiga (Univ. Milano-Bicocca) who wished to list irreducible \({\mathbb C}G\)-modules \(V\) with fixed point free action of \(T\). The proof is not quite straightforward and uses some machinery of representation theory of groups with cyclic Sylow \(p\)-subgroup over \(p\)-adic integers. For \(p\) is used a Zsigmondy prime for \(q^n-1\). The difficulties arise as \(T\) is not a \(p\)-group. In fact, the case where \(n\) is a prime is completed by using standard facts of Deligne-Lusztig theory. This case is viewed as the base of induction on the number of prime divisors of \(n\). Surprisingly, the induction step requires a lot of efforts, and some possibly new ideas. Details will be explained in the talk.