Let be a cuspidal automorphic representation of PGL of arithmetic conductor and archimedean parameter , and let be an -normalized automorphic form in the space of . The sup-norm problem asks for bounds on in terms of and . The quantum unique ergodicity (QUE) problem concerns the limiting behavior of the -mass of . Previous work on these problems in the conductor-aspect has focused on the case that is a newform.
In this work, we study these problems for a class of automorphic forms that are not newforms. Precisely, we assume that for each prime divisor of , the local component is supercuspidal (and satisfies some additional technical hypotheses), and consider automorphic forms for which the local components are "minimal" vectors. Such vectors may be understood as non-archimedean analogues of lowest weight vectors in holomorphic discrete series representations of PGL.
For automorphic forms as above, we prove a sup-norm bound that is sharper than what is known in the newform case. In particular, if is a holomorphic discrete series of lowest weight , we obtain the optimal bound . We prove also that these forms give analytic test vectors for the QUE period, thereby demonstrating the equivalence between the strong QUE and the subconvexity problems for this class of vectors. This finding contrasts the known failure of this equivalence  for newforms of powerful level.
Cite this article
Yueke Hu, Paul D. Nelson, Abhishek Saha, Some analytic aspects of automorphic forms on GL(2) of minimal type. Comment. Math. Helv. 94 (2019), no. 4, pp. 767–801DOI 10.4171/CMH/473