Representations of Special Compact Linear Groups of Order Two : Construction, Representation Growth, Group Algebras and Branching Laws

Abstract

Let o be the ring of integers of a non-Archimedean local field such that the residue field has characteristic p: Let p be the maximal ideal of o: For Char¹oº = 0; let e be the ramification index of o: i.e., 2o = pe: Let GLn¹oº be the group of n n invertible matrices with entries from o and SLn¹oº be the subgroup of GLn¹oº consisting of all determinant one matrices. In this dissertation, our focus is on the construction of the continuous complex irreducible representations of the group SL2¹oº and to describe the representation growth. Also we will discus some results about group algebras of SL2 ¹o pr º for large r and branching laws obtained by restricting irreducible representations of GL2¹o pr º to SL2¹o pr º: Construction: For r 1; the construction of irreducible representations of GL2¹o pr º and for SL2¹o pr º with p > 2 are known by the work of Jaikin-Zapirain and Stasinski-Stevens. However, those methods do not work for p = 2: In this case we give a construction of all irreducible representations of groups SL2¹o pr º; for r 1with Char¹oº = 2 and for r 4e + 2 with Char¹oº = 0: RepresentationGrowth: For a rigid groupG; it iswell knownthat the abscissa of convergence ¹Gº of the representation zeta function ofG gives precise information about its representation growth. Jaikin-Zapirain and Avni-Klopsch-Onn-Voll proved that ¹SL2¹oºº = 1; for either p > 2 or Char¹oº = 0: We complete these results by proving that ¹SL2¹oºº = 1 also for p = 2 and Char¹oº > 0: Group Algebras: The groups GL2¹o pr º and GL2¹ q»t¼ ¹tr ºº need not be isomorphic, but the group algebras »GL2¹o pr º¼ and »GL2¹ q»t¼ ¹tr ºº¼ are known to be isomorphic. In parallel for p > 2 and r 1; the group algebras »SL2¹o pr º¼ and »SL2¹ q»t¼ ¹tr ºº¼ are also isomorphic. We show that for p = 2 and Char¹oº = 0; the group algebras »SL2¹o pr º¼ and »SL2¹ q»t¼ ¹tr ºº¼ are not isomorphic for r 2e + 2: As a corollary we obtain that the group algebras, »SL2¹ 2r º¼ and »SL2¹ 2»t¼ ¹tr ºº¼ are not isomorphic for r 4: Branching Laws: We give a description of the branching laws obtained by restricting irreducible representations of GL2¹o pr º to SL2¹o pr º for p = 2: In this case, we again show that many results for p = 2 are quite different from the case p > 2: