The theory of \(L\)-indistinguishability for inner forms of \(SL_2\) has been established in the well-known paper of Labesse and Langlands (L-indistinguishability for SL\((2)\). Canad. J. Math. 31 (1979), no. 4, 726-785).

In this memoir, the authors study \(L\)-indistinguishability for inner forms of \(SL_n\) for general \(n\). Following the idea of Vogan in (The local Langlands conjecture. Representation theory of groups and algebras, 305-379, Contemp. Math. 145 (1993)), they modify the \(S\)-group and show that such an \(S\)-group fits well in the theory of endoscopy for inner forms of \(SL_n\).

Table of Contents

• Introduction
• Restriction of representations
• Whittaker normalization over local fields
• Restriction of cusp forms
• Whittaker normalization over global fields
• Endoscopy and its automorphisms
• A conjectural formula for endoscopic transfer
• Descent to Levi subgroups
• Relevance conditions for Langlands parameters
• Endoscopy for inner forms of \(GL_n\)
• Local Langlands correspondence for inner forms of \(GL_n\)
• \(L\)-packets for inner forms of \(SL_n\)
• \(L\)-packets for inner forms of \(SL_n\) over Archimedean fields
• Multiplicity formula for \(SL_n\)
• Multiplicity formula for inner forms of \(SL_n\)
• Lemmas for trace formula
• Trace formula
• Transfer factors
• Bibliography