We can also observe that, in view of the ﬁnite dimensionality, only ﬁnitely many of mπ's are nonzero. Combining this with (2.2), we see that the product of any of the matrix coefficients of representationsπ1, π2∈G can be written as a finite linear combination of matrix coeﬃcients of the representations from (2.3) with non-zero Clebsch-Gordan coefficients. As discussed in Section 1.3, the Casimir element of the universal enveloping algebra U(g) can be viewed as an elliptic linear second-order bi-invariant partial differential.

The fundamental result on compact groups is the Peter-Weyl theorem [PW27], which gives a decomposition of L2(G) into eigenspaces of the Laplacian LG onG, which we now outline. The first estimate follows immediately from the asymptotic Weyl formula for the eigenvalue counting function of the elliptic first-order operator (I− LG)1/2 on the compact manifold. more work from the Weyl sign formula, where rankG denotes the rank of G. Different spaces of functions and distributions can be characterized in terms of the Fourier coeﬃcients.

The set Σ can be considered a special case of the direct sum of Hilbert spaces described in (1.29), with the corresponding interpretation in terms of von Neumann algebras. However, much of the general machinery can be simplified in the present setting since the Fourier coecients allow the interpretation of matrices indexed over the discrete set G, with the dimension of each matrix equal to the dimension of the corresponding representation. It can be easily checked that they agree with their restrictions on spaces of test functions, which explains the appearance of the inversion mapping ι.

For a review of the representation theory of compact Lie groups and further constructions using the Littlewood-Paley decomposition based on the heat kernel, we refer to Stein's book [Ste70b].

This can also be seen from the point of view of general eigenfunctional expansions of the function of compact manifolds, see [DR16] for a treatment of more general Komatsu-type classes of ultradifferentiable functions and ultradistributions, drawing on an analogous description for analytic functions by Seeley . [See 69]. This space and the Hausdorf-Young inequality for it become useful, for example, in proving Proposition 2.1.2. Related to these (G)-spaces as spaces of weighted sequences with weights given by powers of dπ, a general theory of interpolation spaces [BL76, Theorem 5.5.1].

However, a version of interpolation theory with change of mass must be used, such as Finally, we establish a relation between the familysp(G) and the corresponding spaces of the Schatten family, which we denote bypsch(G), defined by the norms.

## Pseudo-diﬀerential operators on compact Lie groups

*Symbols and quantization**Diﬀerence operators and symbol classes**Symbolic calculus, ellipticity, hypoellipticity**Fourier multipliers and L p -boundedness**Sharp G˚ arding inequality*

We recall from section 1.3 that Xα denotes the invariant left partial diﬀerential operators of order |α| corresponding to a basis of left-invariant vector fieldsX1,· · ·, Xn,n= dimG, of the Lie algebra. In [RT10a], the relevant notion of difference operators is introduced leading to the symbolic calculus of operators on G. However, we now follow the ideas of [RTW14] with a slightly more general treatment of difference operators.

Let q∈C∞(G) disappear from order k∈Nat the unit elemente∈G, that is Dq)(e) = 0 for all left-invariant differential operators D∈Diﬀk−1(G) of order k−1. We now define families of first-order difference operators that replace derivatives in the frequency variable in the Euclidean setting. The set {qj}nj=1 is strongly admissible, and the corresponding difference operators have the form.

For partial difference operators, it can be easily seen that the application of difference operators reduces the order of symbols. The usual H¨ormander classes Ψm(G) of pseudo-differential operators on G seen as a manifold can be characterized in terms of the matrix-valued symbols. For each left-invariant differential operator D ∈ Diﬀk(G) of order k and each difference operator Δq ∈diﬀl(G) of orderl the symbol estimate.

If ρ > δ, this definition is independent of the choice of a strongly allowed collection of difference operators. Here we fix some strongly admissible collections of difference operators, with corresponding operators Xx(α) coming from the Taylor expansion formula (2.24). We say ata:G×G×G →Σ is amatrix-valued amplitude in the class Amρ,δ(G) if, for a strongly admissible collection of difference operators on G, we have the amplitude inequalities.

Moreover, if 0 ≤δ < ρ≤1, then A = Op(a) is a pseudodifferential operator with matrix-valued symbol σA∈Sρ,δm(G) that has an asymptotic expansion. Let A ∈ Op(Sρ,δm(G)) be a pseudodifferential operator with symbol σA ∈ Sρ,δm(G) which is invertible for all but finitely large π∈G, and for all such π satisfies. The conditions there are formulated using special explicit expressions involving the Clebsch-Gordan coefficients on SU(2), but can be reformulated in a much shorter form using the concept of difference operators.

In order to formulate the results, we need to specify a certain collection of first-order diﬀerence operators associated with the elements of the unitary dual G. They satisfy, for example, the ﬁnite Leibniz formula (while general diﬀerence operators. satisfy only the asymptotic Leibniz formula, see [RT10a, Section 10.7.4]) . Such verification relies heavily on the developed symbolic calculus, Leibniz's rules for difference operators, and criteria for weak (1,1) type in terms of properly defined softeners.

In addition, assumption (2.34) can be further improved: namely, to form a strongly admissible family of first-order difference operators that give Dα in (2.28) and (2.29), it is sufficient to take only πk ∈Θ0, the set of irreducible components of the associated representation.