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It is well-known that (left) $G$-invariant differential operators on a Lie group $G$, has an algebraic description, i.e. universal enveloping algebra of the Lie algebra of the group.

On the other hand $\Psi$Do are generalization of the differential operators on general smooth manifold.

My question is that: Does any algebraic description for the $G$-invariant $\Psi$DOs on the Lie groups (or more general on homogeneous spaces $G/H$) exist ? Thanks

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What is a $\Psi$DO? –  GH from MO Apr 19 '13 at 13:28
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Pseudo differential operator. –  Marc Palm Apr 19 '13 at 13:29
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I figured it out: pseudo-differential operator. Very bad writing. –  GH from MO Apr 19 '13 at 13:30
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@Marc: Thank you. There should be a fine for such abbreviations. –  GH from MO Apr 19 '13 at 13:30
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Who pays the fine for $K$-theory? –  Sönke Hansen Apr 22 '13 at 15:27

5 Answers 5

I have never seen such a characterization (and I'm inclined to agree with André that there may not be one at the same level of simplicity as in the case of invariant differential operators), but my best bet towards such one would be to employ a Beals-Cordes-type characterization of pseudodifferential operators: a bounded operator $T$ in the Hilbert space $L^2(\mathbb{R}^n)$ is a pseudodifferential operator of order zero if and only if its commutators with any element of the Lie algebra generated by the coordinate vector fields and (multiplication by) the linear monomials in $\mathbb{R}^n$ (seen as the Abelian Lie group of $n$-dimensional translations) are bounded operators in $L^2(\mathbb{R}^n)$ as well. Obviously, the invariant operators are those that, in addition, have zero commutator with all elements of the Lie algebra of $\mathbb{R}^n$. See for instance H. O. Cordes, "The Technique of Pseudodifferential Operators" (Cambridge, 1995), Chapter 8.

An appropriate replacement of the above Lie algebra in the case of a (reductive) Lie group would involve its Lie algebra and the analog of multiplication by linear monomials, which probably involves the Fourier-Helgason transform (see, for instance, S. Helgason, "Geometric Analysis on Symmetric Spaces" (AMS, 1994), Chapter III).

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It seems to me that an interesting "first step" or sub-problem---and one that should be fairly straightforward, is to characterize the space of invariant symbols of pseudo differential operators on a Lie group (or more generally on a homogeneous space of a Lie group).

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There is the following description of $G$ invariant pseudodifferential operators on a Riemannian homogeneous space $G/H$: The Schwartz kernels are smooth outside the diagonal and conormal with respect to the diagonal; this is equivalent to Beals' commutator characterization mentioned in Pedro Lauridsen Ribeiro's answer. Moreover, the full geometric symbols are, modulo symbols of order $-\infty$, invariant under the symplectic action of $G$ on $T^*(G/H)$.

Geometric symbols are defined by pulling back Schwartz kernels under the exponential map of the Levi-Civita connection to a neighbourhood of the zero section of the tangent bundle $T(G/H)$, and then taking Fourier transforms in the fiber variable. See here for more details of and references to the geometric pseudodifferential calculus. That principal symbols must be invariant is clear from the transformation behaviour under diffeomorphisms (special case of Egorov's theorem, or FIO calculus). Lemma 6.2 in the paper in Math. Z., of which I am a coauthor, gives the invariance of the full geometric symbol if the diffeomorphism is an isometry, e.g. an action on $G/H$ by a group element.

The description discards smoothing operators. This is reasonable if one adopts, from Sato's microlocal analysis, the view that pseudodifferential operators operate on microfunctions.

The above characterization of invariant pseudodifferential operators is not algebraic. However, I think that it is simple and straightforward. I don't know if there is a characterization of $G$ invariant microdifferential (pseudodifferential) operators in the setting of algebraic analysis which is more algebraic.

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I am tempted to answer your question by the negative:
I believe that there does not exist any other, simpler, way of saying "left-invariant pseudo-differential operator on $G$".

The reason is that if you look at the subset of smoothing operators (operators with smooth integral kernel), I already don't know how to say "left-invariant smoothing operator on $G$" in any simpler (algebraic) way.

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Rather then only considering invariant PSD operators, you might want to consider all $G$ invariant operators, i.e., $G$-intertwiner. I describe their functional calculus below. Their functional calculus can be realized/studied via convolution products and representation theory. What I describe is in the realm of the first answer by Pedro Lauridsen Ribeiro. I let you decide whether this classifies as "algebraic", but it is certainly of operator-algebraic/ representation-theoretic flavour. I claim everything else which is $G$-invariant operator will have a equivalent functional calculus.

Here is an example. Assume $H$ is compact. We identify $L^2(G/H)$ with the induced representation $\pi = Ind_{H}^{G} 1$ or the $H$-invariant vectors in $L^2(G)$ and then uses the convolution operators for $\phi \in C_c^\infty(G//H)$: $$T_\phi f(g) = \int\limits_{G} \phi(x) f(xg) d x.$$ E.g. for $G =SL_2(\mathbb{R})$ and $H=SO(2)$, the algebra $C_c^\infty(G//H)$ is commutative by the Gelfand trick (this is not so essential) and the trace $T_\phi$ is an integral of the Harish-Chandra/Selberg transform of $\phi$ over the spectrum of the hyperbolic Laplacian or, alternatively, an integral $$ \int\limits tr\; \pi(\phi) d_{Pl} \pi$$ over the irreducible unitary (tempered) reps $\pi$ of $G$ with $H$-invariant vectors (only principal series representations here). The measure $d_{Pl}$ is the Plancherel measure.

If $H$ is not compact, you can still do something similar working with $C_c^\infty(G)$ and obtain a similar analysis. E.g. take the two important situations when $H$ is a lattice or a parabolic subgroup in a reductive Lie group. The advantage: this generalizes to locally compact groups. E.g. on reductive groups over non-archimedean fields, there are no differential operators in any obvious way, but this gives you the Hecke operators. These ideas are crucial also in the context of the Selberg trace formula.

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