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Orthogonality is a fundamental theme in representation theory and Fourieranalysis. An orthogonality relation for characters of finite abelian groups(now recognized as an orthogonality relation on GL(1)) was used by Dirichlet toprove infinitely many primes in arithmetic progressions. Asymptoticorthogonality relations for GL$(n)$, with $n\le 3$, and applications to numbertheory, have been considered by various researchers over the last 45 years.Recently, the authors of the present work have derived an explicit asymptoticorthogonality relation, with a power savings error term, for GL$(4,\mathbb R)$.Here we we extend those results to GL$(n,\mathbb R)$ $(n\ge2)$.  For $n\le 5$ our results are unconditional. In particular, the case $n=5$represents a new result. The key new ingredient for the proof of the case $n=5$is the theorem of Kim-Shahidi that functorial products of cusp forms onGL(2)$\times$GL(3) are automorphic on GL(6). For $n>5$ our results areconditional on two conjectures, both of which have been verified in variousspecial cases. The first of these conjectures regards lower bounds forRankin-Selberg L-functions, and the second concerns recurrence relations forMellin transforms of GL$(n,\mathbb R)$ Whittaker functions. Our methods assumethe Ramanujan conjecture at the infinite place for Maass cusp forms, but thisassumption can be removed with a weakening in our error term.  Central to our proof is an application of the Kuznetsov Trace formula, and adetailed analysis, utilizing a number of novel techniques, of the variousentities -- Hecke-Maass cusp forms, Langlands Eisenstein series, sphericalprincipal series Whittaker functions and their Mellin transforms, and so on --that arise in this application.

An Asymptotic Orthogonality Relation for ${\rm GL}(n, \mathbb R)$