Rigid Local Systems and Sporadic Simple Groups

We continue the program set up in [Geom. Funct. Anal. <bold>31</bold> (2021), no. 3, 562–662] to study the monodromy groups of hypergeometric and Kloosterman sheaves. We gave there easy to apply criteria on these sheaves that their monodromy groups satisfy the group-theoretic condition <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis bold upper S plus right-parenthesis"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext mathvariant="bold">S+</mml:mtext> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {(\textbf {S+})}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and showed that many of the finite almost quasisimple groups occur as monodromy groups of such sheaves. Here, we show that precisely 12 of the 26 sporadic simple groups occur in this way (and explain why the others cannot occur this way). We also treat some small rank finite groups of Lie type, as well as certain primitive complex reflection groups.