Monodromy groups of Kloosterman and hypergeometric sheaves

Exponential sums and total Weil representations of finite symplectic and unitary groups

“…Moreover, H j has property (S+) by Proposition 5.10, and by Theorem 10.2, H (∞) j = G n,m,−j geom is the image of SU n (q) in the relevant irreducible Weil representation, with H j /H (∞) j being cyclic of order dividing A. Hence, PSU n (q) is the unique non-abelian composition factor of H j , and by Theorem 8.3 and Corollary 8.4 of [KT4], (10.3.1)…”

“…Moreover, our results lead to hypergeometric sheaves whose geometric monodromy groups are Sp 2n (q) for any n ≥ 2, and the general unitary groups GU n (q) for any odd n ≥ 3. This paper may also be viewed as a companion piece to [KT4], which determines which almost quasisimple groups can possibly occur as monodromy groups of hypergeometric sheaves.…”

“…In the special case max(A, B) = 9, suppose A − B = 7 or 8 (possible only when p = 7, respectively when p = 2). Then for any χ with χ A = 1, H big,A,B,χ satisfies condition (S+) (as defined in [KT4,§1]).…”

“…The ranks are both prime to p, being (q c ± 1)/2. So we may apply [KT4,Theorem 1.5]. We must show that W > (D − 1)/2 for H small,A,B and W > D/2, that D − 1 ≥ 4, and check that neither…”