Yes, it is entirely possible. If collection is not large enough to cause memory issues, you can put it in
Mage::registry(), it will be accessible throughout the life-cycle of that request afterwards.
Also, if both functions are in same class, you can put that collection in a class property, this way, your collection will be available only in that class but to all the methods.
In case, you are trying to retain data between multiple requests, I'd suggest using sessions. As mentioned in the comments, sessions are not an option here. So, you can try that with the frontend only. Collection will not be re-used, but I don't think that is going to be any issue anymore, as collection will not be loaded twice in one request cycle.
To achieve that, you can try mimicking what Magento does with category pages for pagination and sorting. Everytime user selects any option in frontend, you will need to retain that data itself in template, either in JS objects or as properties of some special HTML elements.
Providing a code sample would be tough unless you show what methods you are using in frontend and how they are being handled in backend, as there can be 100 possible ways to get this and my sample could be completely different than what you have now. Some changes in your existing code will be little headache than having a total new approach for some minor issue.
Also, looking at the code you supplied, if it's just about working with the multiple actions in same controller with pre-defined parameters(I highly doubt that's the case though!), wrap your logic inside a method that is not an action, and call that method in both of your actions. You'll get identical collection in both cases.