Induced representationsFrobenius reciprocityA second construction of induced representations.Frobenius reciprocityWe shall use this alternative definition of the induced representationto give a proof of Frobenius reciprocity. We first locate the originalrepresentation of H on F inside the induced module:Proof of Frobenius reciprocitybyWe want to prove thatTo see this observe thatwhich says thatThe map s ® TS is clearly linear. AlsoProof of Frobenius reciprocity.Still more definitions of induced representation.In our construction of G(E) from a representation of H, the sectionswhich vanish except at a point x Î M=G/H are identified with elements of the fiber Ex . So if m = H, and W:= Em, and V:= G(E) then W is a subspace of the G-module V stable under the action of the elements of H and V is the direct sum of the images of W under the left cosets sH. This is the definition in Serre page 28. It suffices to use s which belong to a system of left coset representa-tives R. The group algebra C[G] of a finite group G is the algebra whichhas a basis indexed by elements of G and whose multiplication extendsthat of G. Any G module becomes automatically a C[G] module.If H is a subgroup of G then C[H] is a subalgebra of C[G] and theelements of R form a basis of C[G] considered as a C[H] module. If W is an H module then C[G]Ä C[H] W is the induced module.Induction followed by restriction.• Let H and K be subgroups of G. We want to study the operation of inducing a module from H to G and then restricting to K. • Double cosets: We let KÕH act on G with K acting on the left and H on the right. The orbits for this action are called double cosets. We choose a set S of double coset representatives, which means that G is the disjoint union of the KsH as s ranges over S. For s ÎS we define Hs := sHs-1 ÇK which is a subgroup of K. •If r is a representation of H, we let rs denote the representation of H given by rs(x):= r(s-1xs).•Claim (r-G)¯K = Ås (rs -K).Proof that (r-G)¯K = Å(rs-K).V is the direct sum of xW for x Î G/H. Let V(s) be the sum overx Î KsH. So V is the direct sum of the V(s) and V(s) is invariantunder K. We wish to show that each V(s) is induced from therepresentation rs of Hs on W. The subgroup of K fixing sW is Hs, and V(s) is the direct sum of the images xsW, x Î K/ Hs. The representation of Hs on sW is given by k(sw) = (ks)w = s(s-1 ks)w and so is equivalent to rs by the isomorphism s: W ® sW.Mackey’s irreducibility criterion.This says that r-G is irreducible if and only if(a) r is irreducible and(b) rs and r¯Hs are disjoint for all s Ï H. Let c be the character of r and let f be the character of r-G . Irreducibilityof r-G says that (f, f)G =1. Frobenius recip. says that (f, f)G = (f¯H, c)H .If cs denotes the character of rs then by the previous slide (r-G)¯H = Å rs -H or f¯H= å cs -H. By Frobenius reciprocity, (f¯H, c)H = (å cs -H, c)H = (å cs , c ¯Hs)Hs.Now when s=e, Hs = H and cs = c. Thus the summand corresponding to s =e must equal one, which is condition (a), and all other summandsmust vanish which is condition (b).Representations of semi-direct products.Let G be the semi-direct product of a group H and an abelian groupA which means that every element of G can be written uniquely as ah with a Î A and h Î H. All irreductible representations of Aare one dimensional and form a group X under multipication. Thegroup H acts via conjuation on A and hence acts on X by (sc)(a)= c(s-1as).Let {ci}be a system of representatives of the orbits of H on X andLet Hi be the isotropy group of ci . Let Gi = A Hi . Extend thefunction ci to Gi by setting ci (ah):= ci(a). We haveci (ahbh’)= ci (ahbh-1 hh’)= ci (ahbh-1 )= ci (a) ci (b)= ci (ah) ci (bh’)so this extension gives a one dimensional representation of Gi .Let r be an irreducible representation of Hi . It gives a representationof Gi via the homomorphism of Gi onto Hi . Take rÄ ci and induceup to G. Call this induced representation R(r, ci ). The claim isthat these are all the irreducible representations of G.Proof that R(r, ci) is irreducible.Suppose that s Ï Gi and let Ks := Gi Çs Gi s-1 . Using Mackey’scriterion, it is enough to show that the restriction of rÄci to Ks isdisjoint from the representation (rÄci )s . But already on A These are disjoint since ci s is not equivalent to ci .Notice that R(r, ci) determines r and ci . Indeed, the restriction ofR(r, ci) to A involves only the characters belonging to the H orbit of ci . So the orbit and hence i is determined. Let W be the space of R(r, ci) and let U be the subspace of W whichTransfors under A according to the character ci. This space isstable under Hi and the restriction of Hi to this subspace acts via r.Proof that the R(r, ci) are all the irreducibles. Suppose that s is an irreducible representation of G on W. DecomposeW according to the irreducibles of A: W =Å Wi where Wi is the subspace of W consisting of those w which satisfy aw =ci(a) w, " aÎG.Since h Wi =Wj where h ci= cj we see that the direct sum of thoseWi where the cI belong to a fixed orbit form an invariant subspace. SoThe direct sum is really over the ci belonging to a single orbit, andhence is an induced representation from a subgroup of the form AHi on Wi . By Mackey’s theorem this representation must be irreducibleand is of the form ci Är as desired.The group AHi is called the little group.The irreducibles of D4, or any even dihedral group.Here a1 is areflection and the Ri are rotations through 90, 180, and 270 degrees.The irreducibles of D4, continued.[(bl,v)]=[(b, s(l)v)]The irreducibles of D4, continued.AlsoThe irreducibles of D4, continued.The irreducibles of D4, continued.The irreducibles of D4, concluded.Our results are summarized in the