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Vectors are representations transform under $SO(3)$ Group, 4-vectors are representations transform under $SO(1,3)$ Group, Like wave function in discrete but infinite basis (hilbert space) are some representations which transform under what? what about continuous basis? what are the Groups which form quantum mechanics?

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    $\begingroup$ Vectors are not representations. Vectors are one example of things that transform under representations. You can think of a representation as a collection of matrices, one corresponding to each group element. For example, for each SO(3) rotation, there is a 3x3 rotation matrix which rotates 3-vectors. $\endgroup$ – G. Smith Mar 7 at 18:34
  • $\begingroup$ You might profitably consult WP. $\endgroup$ – Cosmas Zachos Mar 7 at 19:52
  • $\begingroup$ So can u define in general a tensor in 3 dimensions with a SO(3) group , what we call things that transform under representations ? @ – G. Smith $\endgroup$ – Robin Raj Mar 9 at 10:28
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The book you might consult on the subject is a classic one, by H Weyl, The Theory of Groups and Quantum Mechanics, Dover Publications, 1950, ISBN 978-1-163-18343-4.

A continuous (Lie) group SO(3) ~ SU(2) is generated by three generators, and its group elements are exponentials of all linear combinations of said generators. An n-dim representation of the group thus consists of $n\times n$ matrix exponentials acting on n-vectors; n-1 is twice the spin; you mention n =3, but it could be n =2, for a simplest spin 1/2 system. The operators and observables of such a simple spin system are in the space of functions of the generators, so, not just the exponentials: this space is called "the universal enveloping algebra" of the so(3) Lie algebra, and is not a Lie algebra, so it is not closed under commutation. So, in this simplest model, your Hilbert space is two dimensional.

(Again, SO(1,3) has six generators with representations which are not just 4×4 dimensional matrices, but quite varied, as you may read up on elementary group theory books or WP.)

The conventional QM Hilbert space you may well have in mind consists of infinite-dimensional vectors acted upon by infinite-dimensional square matrices in the universal enveloping algebra of the Heisenberg Lie algebra (generated by $\hat X, \hat P$ and the center thereof, the identity matrix in the infinite-dim representation of the continuous Heisenberg group ).

Thus, your operators are functions of the above generators, acting on, e.g., eigenvectors of $\hat X$, an infinite-dimensional vector space, now. There are further technical restrictions such as square integrability, but they apparently outrange the scope of your question.

The Heisenberg group provides suitable changes of basis of such a Hilbert space, and, again, Hamiltonians generate upon exponentiation continuous "rotations" of such Hilbert space states, even though they themselves do not close into the Heisenberg algebra. Instead, the Heisenberg algebra specifies the (unitary) quantum canonical transformations (motion) they (the Hamiltonians) generate, the group Weyl calls that of quantum kinematics.

You may usefully train your intuition as Weyl did in his book, by taking the infinite-dimension limit of finite Hilbert spaces acted upon by Sylvester's (1882) clock and shift matrices.

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First, we need to distinguish between a mathematician's representation and a physicist's. Formally, a representation of a Lie group $G$ on a vector space $V$ is a homomorphism $\rho : G \to \mathrm{Aut}(V)$ such that,

$$\rho(g_1g_2) = \rho(g_1)\rho(g_2).$$

$\rho$ is the representation, but physicists will often call $V$ the representation (space) or elements $v\in V$ as the representation. For example, Euclidean spinors transform under $\mathrm{Spin}(n)$.

Note that $\rho(g)$ need not be a matrix, and the only constraint is that it act linearly on the space, so there are many non-trivial actions one can define.

The Hilbert space in quantum mechanics is a space which - as said in Comas' answer - is acted upon by the Heisenberg group.

However, the wave function is slightly more complicated than say, a vector. Formally, the wave function is a section of an associated bundle to a principal bundle constructed from a symplectic form. The advantage of this is to formulate the dynamics in a coordinate independent way, as for example self-adjointness fails in quantum mechanics for certain choices of coordinates.

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