A strong Gelfand pair \((G, H)\) is a finite group \(G\) together with a subgroup \(H\) such that every irreducible (complex) character of \(H\) induces a multiplicity-free character of \(G\). In other words, for any \(\chi \in \mathrm{Irr}(G)\) and any \(\psi \in \mathrm{Irr}(H)\) we have \([\chi, \psi^G]_G\leq 1\). Equivalently, \((G, H)\) is a strong Gelfand pair if and only if the Schur ring determined by the \(H\)-classes \(g^H = \{h^{-1} g h : h\in H\}\), where \(g \in G\), is commutative.
Extending the results of [A. Burton and S. Humphries, J. Algebra Appl. 22, No. 6, Article ID 2350133, 13 p. (2023; Zbl 1517.20070)], in this paper the authors classify the strong Gelfand pairs \((G,H)\) when \(G=\mathrm{SL}(2,q)\).
In fact, fix the following notation:

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\(C_n\) is the cyclic subgroup of order \(n\);

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\(B=C_q\rtimes C_{q-1}\) is the Borel subgroup of upper triangular matrices;

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\(B_2\leq B\) is the subgroup of \(B\) whose diagonal entries are squares in \(\mathbb{F}_q^\times\);

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\(D_{2(q+1)}\) is the maximal dihedral subgroup of order \(2(q+1)\) (when \(p=2\)).

Then, the authors prove the following result. Let \(q=p^n > 11\), where \(p\) is a prime, and let \(G=\mathrm{SL}(2,q)\). Up to conjugacy, the strong Gelfand pairs \((G,H)\), where \(G\neq H\), are exactly those with:

\((i)\)

\(H=B\) if \(p\equiv 1 \pmod 4\);

\((ii)\)

\(H\in \{B, B_2\}\) if \(p\equiv 3 \pmod 4\);

\((iii)\)

\(H \in \{B, D_{2(q+1)}, C_{q+1}\}\) if \(p=2\).

A full classification is also given for the cases \(q=p\in \{2,3,5,7,9,11\}\).